{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0001","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"74b176b78352471d374220145d451aeaa04f2344f60b002df7a0373c26343181","family_id":"qf-continuity-affine-value","seed":202610083001,"semantic_template":"qf-continuity-affine-value/{\"inputs\":[\"value_b\",\"0\"],\"operation\":\"continuous_affine_value\"}"},"fields":{"title":"Point Value of a Continuous Affine Model","stem":"On the whole real domain, f(x)=(0)x+(0). Find the defined point value f(0).","answer":{"kind":"short_text","value":"0","explanation":"Exact substitution gives m*a+b=0. This is the stated total affine model's point value, not a silently filled value of a punctured function."},"skills":["Point Value of a Continuous Affine Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0002","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b1d3114da310ffe31b12547cfba035a438354633efc769cdefeeb61f5d43bf27","family_id":"qf-continuity-affine-value","seed":202610083002,"semantic_template":"qf-continuity-affine-value/{\"inputs\":[\"value_b\",\"-11\"],\"operation\":\"continuous_affine_value\"}"},"fields":{"title":"Point Value of a Continuous Affine Model","stem":"On the whole real domain, f(x)=(-2)x+(-11). Find the defined point value f(0).","answer":{"kind":"short_text","value":"-11","explanation":"Exact substitution gives m*a+b=-11. This is the stated total affine model's point value, not a silently filled value of a punctured function."},"skills":["Point Value of a Continuous Affine Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0003","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"181644d301a5c24674dbf050c35ca11dd1d475fe30f32262521a0ad2dbdacc46","family_id":"qf-continuity-affine-value","seed":202610083003,"semantic_template":"qf-continuity-affine-value/{\"inputs\":[\"3\",\"-10\",\"3\"],\"operation\":\"continuous_affine_value\"}"},"fields":{"title":"Point Value of a Continuous Affine Model","stem":"On the whole real domain, f(x)=(3)x+(-10). Find the defined point value f(3).","answer":{"kind":"short_text","value":"-1","explanation":"Exact substitution gives m*a+b=-1. This is the stated total affine model's point value, not a silently filled value of a punctured function."},"skills":["Point Value of a Continuous Affine Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0004","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a1ff49bd6d8c06de7f0c2bd22a7397a2a1af17817c9f482ea5ca504135867ab9","family_id":"qf-continuity-affine-value","seed":202610083004,"semantic_template":"qf-continuity-affine-value/{\"inputs\":[\"-4\",\"-9\",\"4\"],\"operation\":\"continuous_affine_value\"}"},"fields":{"title":"Point Value of a Continuous Affine Model","stem":"On the whole real domain, f(x)=(-4)x+(-9). Find the defined point value f(4).","answer":{"kind":"short_text","value":"-25","explanation":"Exact substitution gives m*a+b=-25. This is the stated total affine model's point value, not a silently filled value of a punctured function."},"skills":["Point Value of a Continuous Affine Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0005","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c5e9165f523276e346581c58865a86bd941666c6e0a210cc30b1f955281d8bcf","family_id":"qf-continuity-affine-value","seed":202610083005,"semantic_template":"qf-continuity-affine-value/{\"inputs\":[\"5\",\"-8\",\"1\"],\"operation\":\"continuous_affine_value\"}"},"fields":{"title":"Point Value of a Continuous Affine Model","stem":"On the whole real domain, f(x)=(5)x+(-8). Find the defined point value f(1).","answer":{"kind":"short_text","value":"-3","explanation":"Exact substitution gives m*a+b=-3. This is the stated total affine model's point value, not a silently filled value of a punctured function."},"skills":["Point Value of a Continuous Affine Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0006","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f2779af064f5f6567985e99b4a23cca660d201570c213a20bcb6f3839d1d81f3","family_id":"qf-continuity-affine-value","seed":202610083006,"semantic_template":"qf-continuity-affine-value/{\"inputs\":[\"-6\",\"-7\",\"2\"],\"operation\":\"continuous_affine_value\"}"},"fields":{"title":"Point Value of a Continuous Affine Model","stem":"On the whole real domain, f(x)=(-6)x+(-7). Find the defined point value f(2).","answer":{"kind":"short_text","value":"-19","explanation":"Exact substitution gives m*a+b=-19. This is the stated total affine model's point value, not a silently filled value of a punctured function."},"skills":["Point Value of a Continuous Affine Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0007","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9db0a771a9fcfa6f99895cc63ed5d6fa4d2fcf96a9c20530234e82fef0106e79","family_id":"qf-continuity-affine-value","seed":202610083007,"semantic_template":"qf-continuity-affine-value/{\"inputs\":[\"7\",\"-6\",\"3\"],\"operation\":\"continuous_affine_value\"}"},"fields":{"title":"Point Value of a Continuous Affine Model","stem":"On the whole real domain, f(x)=(7)x+(-6). Find the defined point value f(3).","answer":{"kind":"short_text","value":"15","explanation":"Exact substitution gives m*a+b=15. This is the stated total affine model's point value, not a silently filled value of a punctured function."},"skills":["Point Value of a Continuous Affine Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0008","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"62483c530059bd6ce64665b6523b49cf1eca95029fde93020fbea01ea4bb5145","family_id":"qf-continuity-affine-value","seed":202610083008,"semantic_template":"qf-continuity-affine-value/{\"inputs\":[\"-8\",\"-5\",\"4\"],\"operation\":\"continuous_affine_value\"}"},"fields":{"title":"Point Value of a Continuous Affine Model","stem":"On the whole real domain, f(x)=(-8)x+(-5). Find the defined point value f(4).","answer":{"kind":"short_text","value":"-37","explanation":"Exact substitution gives m*a+b=-37. This is the stated total affine model's point value, not a silently filled value of a punctured function."},"skills":["Point Value of a Continuous Affine Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0009","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ad56374cd77842bc5be3becd3d7d553b9f1f1b0d73c6c794c2f7d244b745470","family_id":"qf-continuity-affine-value","seed":202610083009,"semantic_template":"qf-continuity-affine-value/{\"inputs\":[\"9\",\"-4\",\"1\"],\"operation\":\"continuous_affine_value\"}"},"fields":{"title":"Point Value of a Continuous Affine Model","stem":"On the whole real domain, f(x)=(9)x+(-4). Find the defined point value f(1).","answer":{"kind":"short_text","value":"5","explanation":"Exact substitution gives m*a+b=5. This is the stated total affine model's point value, not a silently filled value of a punctured function."},"skills":["Point Value of a Continuous Affine Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0010","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c594dd706200cc04dfd742f91d92d5595f3d41f7a8c5011271f0d5c3b52c8e92","family_id":"qf-continuity-affine-value","seed":202610083010,"semantic_template":"qf-continuity-affine-value/{\"inputs\":[\"-10\",\"-3\",\"2\"],\"operation\":\"continuous_affine_value\"}"},"fields":{"title":"Point Value of a Continuous Affine Model","stem":"On the whole real domain, f(x)=(-10)x+(-3). Find the defined point value f(2).","answer":{"kind":"short_text","value":"-23","explanation":"Exact substitution gives m*a+b=-23. This is the stated total affine model's point value, not a silently filled value of a punctured function."},"skills":["Point Value of a Continuous Affine Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0011","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9757302a3f2044755ab6d9cebbb5f649bc72e67dcb1f4735e75367eac32ff5b4","family_id":"qf-continuity-affine-value","seed":202610083011,"semantic_template":"qf-continuity-affine-value/{\"inputs\":[\"11\",\"-2\",\"3\"],\"operation\":\"continuous_affine_value\"}"},"fields":{"title":"Point Value of a Continuous Affine Model","stem":"On the whole real domain, f(x)=(11)x+(-2). Find the defined point value f(3).","answer":{"kind":"short_text","value":"31","explanation":"Exact substitution gives m*a+b=31. This is the stated total affine model's point value, not a silently filled value of a punctured function."},"skills":["Point Value of a Continuous Affine Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0012","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"76e57c37c02c6efc4f85c34f008bd3b6f571ce716eb363c61dc9dc3aedcf418c","family_id":"qf-continuity-affine-value","seed":202610083012,"semantic_template":"qf-continuity-affine-value/{\"inputs\":[\"-12\",\"-1\",\"4\"],\"operation\":\"continuous_affine_value\"}"},"fields":{"title":"Point Value of a Continuous Affine Model","stem":"On the whole real domain, f(x)=(-12)x+(-1). Find the defined point value f(4).","answer":{"kind":"short_text","value":"-49","explanation":"Exact substitution gives m*a+b=-49. This is the stated total affine model's point value, not a silently filled value of a punctured function."},"skills":["Point Value of a Continuous Affine Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0013","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"739f3e9ac88d28876bea4ba1022051946cb8f88faef1e7d974a9368efc6125c1","family_id":"qf-continuity-affine-value","seed":202610083013,"semantic_template":"qf-continuity-affine-value/{\"inputs\":[\"13\",\"0\",\"1\"],\"operation\":\"continuous_affine_value\"}"},"fields":{"title":"Point Value of a Continuous Affine Model","stem":"On the whole real domain, f(x)=(13)x+(0). Find the defined point value f(1).","answer":{"kind":"short_text","value":"13","explanation":"Exact substitution gives m*a+b=13. This is the stated total affine model's point value, not a silently filled value of a punctured function."},"skills":["Point Value of a Continuous Affine Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0014","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"26e92453ce1fb7570e3d15f437728ff485dcdb8554bd2dec2e63578f6da4eb65","family_id":"qf-continuity-affine-value","seed":202610083014,"semantic_template":"qf-continuity-affine-value/{\"inputs\":[\"-14\",\"1\",\"2\"],\"operation\":\"continuous_affine_value\"}"},"fields":{"title":"Point Value of a Continuous Affine Model","stem":"On the whole real domain, f(x)=(-14)x+(1). Find the defined point value f(2).","answer":{"kind":"short_text","value":"-27","explanation":"Exact substitution gives m*a+b=-27. This is the stated total affine model's point value, not a silently filled value of a punctured function."},"skills":["Point Value of a Continuous Affine Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0015","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d351b5557df886c08807178c473fdaa13851858bc2c8b6e1abd06f8732a88ab5","family_id":"qf-continuity-affine-value","seed":202610083015,"semantic_template":"qf-continuity-affine-value/{\"inputs\":[\"15\",\"2\",\"3\"],\"operation\":\"continuous_affine_value\"}"},"fields":{"title":"Point Value of a Continuous Affine Model","stem":"On the whole real domain, f(x)=(15)x+(2). Find the defined point value f(3).","answer":{"kind":"short_text","value":"47","explanation":"Exact substitution gives m*a+b=47. This is the stated total affine model's point value, not a silently filled value of a punctured function."},"skills":["Point Value of a Continuous Affine Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0016","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"24b795f1a61832bdb3f60585aa2a39d2480f3b428e7babe86d8b3236b74e6a39","family_id":"qf-continuity-affine-value","seed":202610083016,"semantic_template":"qf-continuity-affine-value/{\"inputs\":[\"-16\",\"3\",\"4\"],\"operation\":\"continuous_affine_value\"}"},"fields":{"title":"Point Value of a Continuous Affine Model","stem":"On the whole real domain, f(x)=(-16)x+(3). Find the defined point value f(4).","answer":{"kind":"short_text","value":"-61","explanation":"Exact substitution gives m*a+b=-61. This is the stated total affine model's point value, not a silently filled value of a punctured function."},"skills":["Point Value of a Continuous Affine Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0017","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3a3e252e6439f0c403141fe6dcee3aa5f7505bb3caf553acd2e35b681a09f778","family_id":"qf-continuity-affine-value","seed":202610083017,"semantic_template":"qf-continuity-affine-value/{\"inputs\":[\"17\",\"4\",\"1\"],\"operation\":\"continuous_affine_value\"}"},"fields":{"title":"Point Value of a Continuous Affine Model","stem":"On the whole real domain, f(x)=(17)x+(4). Find the defined point value f(1).","answer":{"kind":"short_text","value":"21","explanation":"Exact substitution gives m*a+b=21. This is the stated total affine model's point value, not a silently filled value of a punctured function."},"skills":["Point Value of a Continuous Affine Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0018","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"761c877a3fb2b7329df8f8aa0bf07e1d7c953353f0ac4120e6a7456473d0480c","family_id":"qf-continuity-affine-value","seed":202610083018,"semantic_template":"qf-continuity-affine-value/{\"inputs\":[\"-18\",\"5\",\"2\"],\"operation\":\"continuous_affine_value\"}"},"fields":{"title":"Point Value of a Continuous Affine Model","stem":"On the whole real domain, f(x)=(-18)x+(5). Find the defined point value f(2).","answer":{"kind":"short_text","value":"-31","explanation":"Exact substitution gives m*a+b=-31. This is the stated total affine model's point value, not a silently filled value of a punctured function."},"skills":["Point Value of a Continuous Affine Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0019","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b1117d1ca6126b2a3f119f090bc66e00766b97ce18b54f89efde8bb99ee9bc67","family_id":"qf-continuity-affine-value","seed":202610083019,"semantic_template":"qf-continuity-affine-value/{\"inputs\":[\"19\",\"6\",\"3\"],\"operation\":\"continuous_affine_value\"}"},"fields":{"title":"Point Value of a Continuous Affine Model","stem":"On the whole real domain, f(x)=(19)x+(6). Find the defined point value f(3).","answer":{"kind":"short_text","value":"63","explanation":"Exact substitution gives m*a+b=63. This is the stated total affine model's point value, not a silently filled value of a punctured function."},"skills":["Point Value of a Continuous Affine Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0020","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d88e068d5472bae5b7699456edf66b7037eab23ce72c94503621d17f05a0845","family_id":"qf-continuity-affine-value","seed":202610083020,"semantic_template":"qf-continuity-affine-value/{\"inputs\":[\"-20\",\"7\",\"4\"],\"operation\":\"continuous_affine_value\"}"},"fields":{"title":"Point Value of a Continuous Affine Model","stem":"On the whole real domain, f(x)=(-20)x+(7). Find the defined point value f(4).","answer":{"kind":"short_text","value":"-73","explanation":"Exact substitution gives m*a+b=-73. This is the stated total affine model's point value, not a silently filled value of a punctured function."},"skills":["Point Value of a Continuous Affine Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0021","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8062e58657a0cf545a38b5a6150cd667dc35e476728ab8df70d5c90a7a36c5fc","family_id":"qf-continuity-affine-value","seed":202610083021,"semantic_template":"qf-continuity-affine-value/{\"inputs\":[\"21\",\"8\",\"1\"],\"operation\":\"continuous_affine_value\"}"},"fields":{"title":"Point Value of a Continuous Affine Model","stem":"On the whole real domain, f(x)=(21)x+(8). Find the defined point value f(1).","answer":{"kind":"short_text","value":"29","explanation":"Exact substitution gives m*a+b=29. This is the stated total affine model's point value, not a silently filled value of a punctured function."},"skills":["Point Value of a Continuous Affine Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0022","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba1c1cf528699e707601062c5835f6accf90f2dd5bae25bb62951cdbd9b20e8c","family_id":"qf-continuity-affine-value","seed":202610083022,"semantic_template":"qf-continuity-affine-value/{\"inputs\":[\"-22\",\"9\",\"2\"],\"operation\":\"continuous_affine_value\"}"},"fields":{"title":"Point Value of a Continuous Affine Model","stem":"On the whole real domain, f(x)=(-22)x+(9). Find the defined point value f(2).","answer":{"kind":"short_text","value":"-35","explanation":"Exact substitution gives m*a+b=-35. This is the stated total affine model's point value, not a silently filled value of a punctured function."},"skills":["Point Value of a Continuous Affine Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0023","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f4aa54fead5443047ed9fbc88383683babb66a75e221b5bf02af406da2bc9109","family_id":"qf-continuity-affine-value","seed":202610083023,"semantic_template":"qf-continuity-affine-value/{\"inputs\":[\"23\",\"10\",\"3\"],\"operation\":\"continuous_affine_value\"}"},"fields":{"title":"Point Value of a Continuous Affine Model","stem":"On the whole real domain, f(x)=(23)x+(10). Find the defined point value f(3).","answer":{"kind":"short_text","value":"79","explanation":"Exact substitution gives m*a+b=79. This is the stated total affine model's point value, not a silently filled value of a punctured function."},"skills":["Point Value of a Continuous Affine Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0024","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c7d79d212bf886275007e0c6d8de9552b4c9a196927cfdc384a16128afee8125","family_id":"qf-continuity-affine-value","seed":202610083024,"semantic_template":"qf-continuity-affine-value/{\"inputs\":[\"-24\",\"11\",\"4\"],\"operation\":\"continuous_affine_value\"}"},"fields":{"title":"Point Value of a Continuous Affine Model","stem":"On the whole real domain, f(x)=(-24)x+(11). Find the defined point value f(4).","answer":{"kind":"short_text","value":"-85","explanation":"Exact substitution gives m*a+b=-85. This is the stated total affine model's point value, not a silently filled value of a punctured function."},"skills":["Point Value of a Continuous Affine Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0025","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec0a9e0d3919affc55cab803b128a9da2ff48393ba9f4ef2c889be142023cb0d","family_id":"qf-continuity-affine-value","seed":202610083025,"semantic_template":"qf-continuity-affine-value/{\"inputs\":[\"25\",\"12\",\"1\"],\"operation\":\"continuous_affine_value\"}"},"fields":{"title":"Point Value of a Continuous Affine Model","stem":"On the whole real domain, f(x)=(25)x+(12). Find the defined point value f(1).","answer":{"kind":"short_text","value":"37","explanation":"Exact substitution gives m*a+b=37. This is the stated total affine model's point value, not a silently filled value of a punctured function."},"skills":["Point Value of a Continuous Affine Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0026","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3110f47e8d8e8302d39865112e0332da847e3c21a206d0f90bb0c491b827f5e9","family_id":"qf-continuity-piecewise-point","seed":202610083026,"semantic_template":"qf-continuity-piecewise-point/{\"inputs\":[\"-13\",\"-13\",\"-13\"],\"operation\":\"piecewise_continuity\"}"},"fields":{"title":"Continuity with Two Sides and a Defined Center Value","stem":"The function is defined on all R. The left branch at x<-12 is (1)x+(-1); the right branch at x>-12 is (-1)x+(-25). Its explicit center value is f(-12)=-13. Is it continuous at -12?","answer":{"kind":"short_text","value":"Yes","explanation":"The side limits are [-13, -13], and the center value is -13. All three must agree, giving Yes; agreement of the sides alone is insufficient."},"skills":["Continuity with Two Sides and a Defined Center Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0027","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c469535f64f2af376412fd776ab17aa92b771a1686886b9216cbac84f5599f98","family_id":"qf-continuity-piecewise-point","seed":202610083027,"semantic_template":"qf-continuity-piecewise-point/{\"inputs\":[\"2\",\"2\",\"3\"],\"operation\":\"piecewise_continuity\"}"},"fields":{"title":"Continuity with Two Sides and a Defined Center Value","stem":"The function is defined on all R. The left branch at x<-11 is (2)x+(24); the right branch at x>-11 is (-2)x+(-20). Its explicit center value is f(-11)=3. Is it continuous at -11?","answer":{"kind":"short_text","value":"No","explanation":"The side limits are [2, 2], and the center value is 3. All three must agree, giving No; agreement of the sides alone is insufficient."},"skills":["Continuity with Two Sides and a Defined Center Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0028","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f9d2ea8815d490a4b909cbc261182362c9ac679998fd9741ac903435fcabea14","family_id":"qf-continuity-piecewise-point","seed":202610083028,"semantic_template":"qf-continuity-piecewise-point/{\"inputs\":[\"-1\",\"1\",\"0\"],\"operation\":\"piecewise_continuity\"}"},"fields":{"title":"Continuity with Two Sides and a Defined Center Value","stem":"The function is defined on all R. The left branch at x<-10 is (3)x+(29); the right branch at x>-10 is (-3)x+(-29). Its explicit center value is f(-10)=0. Is it continuous at -10?","answer":{"kind":"short_text","value":"No","explanation":"The side limits are [-1, 1], and the center value is 0. All three must agree, giving No; agreement of the sides alone is insufficient."},"skills":["Continuity with Two Sides and a Defined Center Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0029","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d741790a3ae4e6dbc4acd8ab6abbcfbd050372654752c558fc6fede3b4b81204","family_id":"qf-continuity-piecewise-point","seed":202610083029,"semantic_template":"qf-continuity-piecewise-point/{\"inputs\":[\"-10\",\"-10\",\"-10\"],\"operation\":\"piecewise_continuity\"}"},"fields":{"title":"Continuity with Two Sides and a Defined Center Value","stem":"The function is defined on all R. The left branch at x<-9 is (4)x+(26); the right branch at x>-9 is (-1)x+(-19). Its explicit center value is f(-9)=-10. Is it continuous at -9?","answer":{"kind":"short_text","value":"Yes","explanation":"The side limits are [-10, -10], and the center value is -10. All three must agree, giving Yes; agreement of the sides alone is insufficient."},"skills":["Continuity with Two Sides and a Defined Center Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0030","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d7379658623861537b0e8c3d60b9e25990f058af6bd4770bd78d5ea6f642a5fa","family_id":"qf-continuity-piecewise-point","seed":202610083030,"semantic_template":"qf-continuity-piecewise-point/{\"inputs\":[\"-8\",\"-8\",\"-9\"],\"operation\":\"piecewise_continuity\"}"},"fields":{"title":"Continuity with Two Sides and a Defined Center Value","stem":"The function is defined on all R. The left branch at x<-8 is (1)x+(0); the right branch at x>-8 is (-2)x+(-24). Its explicit center value is f(-8)=-9. Is it continuous at -8?","answer":{"kind":"short_text","value":"No","explanation":"The side limits are [-8, -8], and the center value is -9. All three must agree, giving No; agreement of the sides alone is insufficient."},"skills":["Continuity with Two Sides and a Defined Center Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0031","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5a6bd463ce09466ccb3cb9601a9c6a1762c163f54cbfd2318a2035d4c16f6388","family_id":"qf-continuity-piecewise-point","seed":202610083031,"semantic_template":"qf-continuity-piecewise-point/{\"inputs\":[\"-8\",\"-6\",\"-8\"],\"operation\":\"piecewise_continuity\"}"},"fields":{"title":"Continuity with Two Sides and a Defined Center Value","stem":"The function is defined on all R. The left branch at x<-7 is (2)x+(6); the right branch at x>-7 is (-3)x+(-27). Its explicit center value is f(-7)=-8. Is it continuous at -7?","answer":{"kind":"short_text","value":"No","explanation":"The side limits are [-8, -6], and the center value is -8. All three must agree, giving No; agreement of the sides alone is insufficient."},"skills":["Continuity with Two Sides and a Defined Center Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0032","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6285ab5ed72aeec1c72c529fa9234c57dfa636217a3a1f7d7c924c5ba6fdae8c","family_id":"qf-continuity-piecewise-point","seed":202610083032,"semantic_template":"qf-continuity-piecewise-point/{\"inputs\":[\"-7\",\"-7\",\"-7\"],\"operation\":\"piecewise_continuity\"}"},"fields":{"title":"Continuity with Two Sides and a Defined Center Value","stem":"The function is defined on all R. The left branch at x<-6 is (3)x+(11); the right branch at x>-6 is (-1)x+(-13). Its explicit center value is f(-6)=-7. Is it continuous at -6?","answer":{"kind":"short_text","value":"Yes","explanation":"The side limits are [-7, -7], and the center value is -7. All three must agree, giving Yes; agreement of the sides alone is insufficient."},"skills":["Continuity with Two Sides and a Defined Center Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0033","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"79f49c415efb5da966bf702e04dd57ac7416c0e17639ed76098db560ef8b1b83","family_id":"qf-continuity-piecewise-point","seed":202610083033,"semantic_template":"qf-continuity-piecewise-point/{\"inputs\":[\"-5\",\"-5\",\"-6\"],\"operation\":\"piecewise_continuity\"}"},"fields":{"title":"Continuity with Two Sides and a Defined Center Value","stem":"The function is defined on all R. The left branch at x<-5 is (4)x+(15); the right branch at x>-5 is (-2)x+(-15). Its explicit center value is f(-5)=-6. Is it continuous at -5?","answer":{"kind":"short_text","value":"No","explanation":"The side limits are [-5, -5], and the center value is -6. All three must agree, giving No; agreement of the sides alone is insufficient."},"skills":["Continuity with Two Sides and a Defined Center Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0034","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b671e2d840f9ab899f11831b7fb736f112f2c51ba184f331e97bff73639ba855","family_id":"qf-continuity-piecewise-point","seed":202610083034,"semantic_template":"qf-continuity-piecewise-point/{\"inputs\":[\"-5\",\"-3\",\"-5\"],\"operation\":\"piecewise_continuity\"}"},"fields":{"title":"Continuity with Two Sides and a Defined Center Value","stem":"The function is defined on all R. The left branch at x<-4 is (1)x+(-1); the right branch at x>-4 is (-3)x+(-15). Its explicit center value is f(-4)=-5. Is it continuous at -4?","answer":{"kind":"short_text","value":"No","explanation":"The side limits are [-5, -3], and the center value is -5. All three must agree, giving No; agreement of the sides alone is insufficient."},"skills":["Continuity with Two Sides and a Defined Center Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0035","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a43ec5d5cb29bd02746149833ac9f289b6dd9933b1561daf108adfe1fd2be7db","family_id":"qf-continuity-piecewise-point","seed":202610083035,"semantic_template":"qf-continuity-piecewise-point/{\"inputs\":[\"-4\",\"-4\",\"-4\"],\"operation\":\"piecewise_continuity\"}"},"fields":{"title":"Continuity with Two Sides and a Defined Center Value","stem":"The function is defined on all R. The left branch at x<-3 is (2)x+(2); the right branch at x>-3 is (-1)x+(-7). Its explicit center value is f(-3)=-4. Is it continuous at -3?","answer":{"kind":"short_text","value":"Yes","explanation":"The side limits are [-4, -4], and the center value is -4. All three must agree, giving Yes; agreement of the sides alone is insufficient."},"skills":["Continuity with Two Sides and a Defined Center Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0036","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d8174ad467cb172759141ded12f44dbaa5a7b664536b1053ab030e39e953ac73","family_id":"qf-continuity-piecewise-point","seed":202610083036,"semantic_template":"qf-continuity-piecewise-point/{\"inputs\":[\"-2\",\"-2\",\"-3\"],\"operation\":\"piecewise_continuity\"}"},"fields":{"title":"Continuity with Two Sides and a Defined Center Value","stem":"The function is defined on all R. The left branch at x<-2 is (3)x+(4); the right branch at x>-2 is (-2)x+(-6). Its explicit center value is f(-2)=-3. Is it continuous at -2?","answer":{"kind":"short_text","value":"No","explanation":"The side limits are [-2, -2], and the center value is -3. All three must agree, giving No; agreement of the sides alone is insufficient."},"skills":["Continuity with Two Sides and a Defined Center Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0037","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"006de99c0e89bacc26ce07a46e3a22236f690a1a42b5c3f991a1792f928bec4d","family_id":"qf-continuity-piecewise-point","seed":202610083037,"semantic_template":"qf-continuity-piecewise-point/{\"inputs\":[\"-2\",\"0\",\"-2\"],\"operation\":\"piecewise_continuity\"}"},"fields":{"title":"Continuity with Two Sides and a Defined Center Value","stem":"The function is defined on all R. The left branch at x<-1 is (4)x+(2); the right branch at x>-1 is (-3)x+(-3). Its explicit center value is f(-1)=-2. Is it continuous at -1?","answer":{"kind":"short_text","value":"No","explanation":"The side limits are [-2, 0], and the center value is -2. All three must agree, giving No; agreement of the sides alone is insufficient."},"skills":["Continuity with Two Sides and a Defined Center Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0038","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"efe161f23db6fdd5ee1a057fd0021d36ff77e5553248cd915d9069adb20decf5","family_id":"qf-continuity-piecewise-point","seed":202610083038,"semantic_template":"qf-continuity-piecewise-point/{\"inputs\":[\"-1\",\"-1\",\"-1\"],\"operation\":\"piecewise_continuity\"}"},"fields":{"title":"Continuity with Two Sides and a Defined Center Value","stem":"The function is defined on all R. The left branch at x<0 is (1)x+(-1); the right branch at x>0 is (-1)x+(-1). Its explicit center value is f(0)=-1. Is it continuous at 0?","answer":{"kind":"short_text","value":"Yes","explanation":"The side limits are [-1, -1], and the center value is -1. All three must agree, giving Yes; agreement of the sides alone is insufficient."},"skills":["Continuity with Two Sides and a Defined Center Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0039","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a46a85d94c57bb987712f760ab71e824f932d4aebdcf1505a6172b94fadf47a7","family_id":"qf-continuity-piecewise-point","seed":202610083039,"semantic_template":"qf-continuity-piecewise-point/{\"inputs\":[\"1\",\"1\",\"0\"],\"operation\":\"piecewise_continuity\"}"},"fields":{"title":"Continuity with Two Sides and a Defined Center Value","stem":"The function is defined on all R. The left branch at x<1 is (2)x+(-1); the right branch at x>1 is (-2)x+(3). Its explicit center value is f(1)=0. Is it continuous at 1?","answer":{"kind":"short_text","value":"No","explanation":"The side limits are [1, 1], and the center value is 0. All three must agree, giving No; agreement of the sides alone is insufficient."},"skills":["Continuity with Two Sides and a Defined Center Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0040","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a114e50c8a6acd10c0b78a3188c38419b61d5d16a4488a8e8ea6e5fa2daef434","family_id":"qf-continuity-piecewise-point","seed":202610083040,"semantic_template":"qf-continuity-piecewise-point/{\"inputs\":[\"1\",\"3\",\"1\"],\"operation\":\"piecewise_continuity\"}"},"fields":{"title":"Continuity with Two Sides and a Defined Center Value","stem":"The function is defined on all R. The left branch at x<2 is (3)x+(-5); the right branch at x>2 is (-3)x+(9). Its explicit center value is f(2)=1. Is it continuous at 2?","answer":{"kind":"short_text","value":"No","explanation":"The side limits are [1, 3], and the center value is 1. All three must agree, giving No; agreement of the sides alone is insufficient."},"skills":["Continuity with Two Sides and a Defined Center Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0041","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d2d7b73c870eed5a0f8da3b0e65b051d184f990dde40b9e4e74305dbd883609","family_id":"qf-continuity-piecewise-point","seed":202610083041,"semantic_template":"qf-continuity-piecewise-point/{\"inputs\":[\"2\",\"2\",\"2\"],\"operation\":\"piecewise_continuity\"}"},"fields":{"title":"Continuity with Two Sides and a Defined Center Value","stem":"The function is defined on all R. The left branch at x<3 is (4)x+(-10); the right branch at x>3 is (-1)x+(5). Its explicit center value is f(3)=2. Is it continuous at 3?","answer":{"kind":"short_text","value":"Yes","explanation":"The side limits are [2, 2], and the center value is 2. All three must agree, giving Yes; agreement of the sides alone is insufficient."},"skills":["Continuity with Two Sides and a Defined Center Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0042","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a0570cf71f46857a057f892f62b01ce584214afae9655c6892af4c54ad17d231","family_id":"qf-continuity-piecewise-point","seed":202610083042,"semantic_template":"qf-continuity-piecewise-point/{\"inputs\":[\"4\",\"4\",\"3\"],\"operation\":\"piecewise_continuity\"}"},"fields":{"title":"Continuity with Two Sides and a Defined Center Value","stem":"The function is defined on all R. The left branch at x<4 is (1)x+(0); the right branch at x>4 is (-2)x+(12). Its explicit center value is f(4)=3. Is it continuous at 4?","answer":{"kind":"short_text","value":"No","explanation":"The side limits are [4, 4], and the center value is 3. All three must agree, giving No; agreement of the sides alone is insufficient."},"skills":["Continuity with Two Sides and a Defined Center Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0043","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ec449b4890e5aa6a848c75b99b29790b72944730da01e9a4bcac30c45889ede","family_id":"qf-continuity-piecewise-point","seed":202610083043,"semantic_template":"qf-continuity-piecewise-point/{\"inputs\":[\"4\",\"6\",\"4\"],\"operation\":\"piecewise_continuity\"}"},"fields":{"title":"Continuity with Two Sides and a Defined Center Value","stem":"The function is defined on all R. The left branch at x<5 is (2)x+(-6); the right branch at x>5 is (-3)x+(21). Its explicit center value is f(5)=4. Is it continuous at 5?","answer":{"kind":"short_text","value":"No","explanation":"The side limits are [4, 6], and the center value is 4. All three must agree, giving No; agreement of the sides alone is insufficient."},"skills":["Continuity with Two Sides and a Defined Center Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0044","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e36163ea8e9ab7cac01d8e32a0dfdcaa070967bc239eeb325ae258b468641d8a","family_id":"qf-continuity-piecewise-point","seed":202610083044,"semantic_template":"qf-continuity-piecewise-point/{\"inputs\":[\"5\",\"5\",\"5\"],\"operation\":\"piecewise_continuity\"}"},"fields":{"title":"Continuity with Two Sides and a Defined Center Value","stem":"The function is defined on all R. The left branch at x<6 is (3)x+(-13); the right branch at x>6 is (-1)x+(11). Its explicit center value is f(6)=5. Is it continuous at 6?","answer":{"kind":"short_text","value":"Yes","explanation":"The side limits are [5, 5], and the center value is 5. All three must agree, giving Yes; agreement of the sides alone is insufficient."},"skills":["Continuity with Two Sides and a Defined Center Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0045","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b49838e754b7bbe73fd42f4ce3a222e6e63a224065f606e61f6fe09ed17b72e2","family_id":"qf-continuity-piecewise-point","seed":202610083045,"semantic_template":"qf-continuity-piecewise-point/{\"inputs\":[\"7\",\"7\",\"6\"],\"operation\":\"piecewise_continuity\"}"},"fields":{"title":"Continuity with Two Sides and a Defined Center Value","stem":"The function is defined on all R. The left branch at x<7 is (4)x+(-21); the right branch at x>7 is (-2)x+(21). Its explicit center value is f(7)=6. Is it continuous at 7?","answer":{"kind":"short_text","value":"No","explanation":"The side limits are [7, 7], and the center value is 6. All three must agree, giving No; agreement of the sides alone is insufficient."},"skills":["Continuity with Two Sides and a Defined Center Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0046","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"37dec9723fd32ab57191a468a18a8c4e277493f22722ae9e4ef927ea2d080bf6","family_id":"qf-continuity-piecewise-point","seed":202610083046,"semantic_template":"qf-continuity-piecewise-point/{\"inputs\":[\"7\",\"9\",\"7\"],\"operation\":\"piecewise_continuity\"}"},"fields":{"title":"Continuity with Two Sides and a Defined Center Value","stem":"The function is defined on all R. The left branch at x<8 is (1)x+(-1); the right branch at x>8 is (-3)x+(33). Its explicit center value is f(8)=7. Is it continuous at 8?","answer":{"kind":"short_text","value":"No","explanation":"The side limits are [7, 9], and the center value is 7. All three must agree, giving No; agreement of the sides alone is insufficient."},"skills":["Continuity with Two Sides and a Defined Center Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0047","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b8e462c0e11ff6b6f793b81cdf06f8410160c15ed362fd58bf027ddc195c5914","family_id":"qf-continuity-piecewise-point","seed":202610083047,"semantic_template":"qf-continuity-piecewise-point/{\"inputs\":[\"8\",\"8\",\"8\"],\"operation\":\"piecewise_continuity\"}"},"fields":{"title":"Continuity with Two Sides and a Defined Center Value","stem":"The function is defined on all R. The left branch at x<9 is (2)x+(-10); the right branch at x>9 is (-1)x+(17). Its explicit center value is f(9)=8. Is it continuous at 9?","answer":{"kind":"short_text","value":"Yes","explanation":"The side limits are [8, 8], and the center value is 8. All three must agree, giving Yes; agreement of the sides alone is insufficient."},"skills":["Continuity with Two Sides and a Defined Center Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0048","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d4de4979731362a9b886a08c1b16fa0060c57fe7aca3f7e88a3fbab6015b9b46","family_id":"qf-continuity-piecewise-point","seed":202610083048,"semantic_template":"qf-continuity-piecewise-point/{\"inputs\":[\"10\",\"10\",\"9\"],\"operation\":\"piecewise_continuity\"}"},"fields":{"title":"Continuity with Two Sides and a Defined Center Value","stem":"The function is defined on all R. The left branch at x<10 is (3)x+(-20); the right branch at x>10 is (-2)x+(30). Its explicit center value is f(10)=9. Is it continuous at 10?","answer":{"kind":"short_text","value":"No","explanation":"The side limits are [10, 10], and the center value is 9. All three must agree, giving No; agreement of the sides alone is insufficient."},"skills":["Continuity with Two Sides and a Defined Center Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0049","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"406e8831c50634418f569594312c8d100be9cd53c8cc485859cc92b8e4fd4b90","family_id":"qf-continuity-piecewise-point","seed":202610083049,"semantic_template":"qf-continuity-piecewise-point/{\"inputs\":[\"10\",\"12\",\"10\"],\"operation\":\"piecewise_continuity\"}"},"fields":{"title":"Continuity with Two Sides and a Defined Center Value","stem":"The function is defined on all R. The left branch at x<11 is (4)x+(-34); the right branch at x>11 is (-3)x+(45). Its explicit center value is f(11)=10. Is it continuous at 11?","answer":{"kind":"short_text","value":"No","explanation":"The side limits are [10, 12], and the center value is 10. All three must agree, giving No; agreement of the sides alone is insufficient."},"skills":["Continuity with Two Sides and a Defined Center Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0050","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"95f77428c14e339f3d1c510a5386957146b2ff5918b7b9bb2ba4ee966aecd78d","family_id":"qf-continuity-piecewise-point","seed":202610083050,"semantic_template":"qf-continuity-piecewise-point/{\"inputs\":[\"11\",\"11\",\"11\"],\"operation\":\"piecewise_continuity\"}"},"fields":{"title":"Continuity with Two Sides and a Defined Center Value","stem":"The function is defined on all R. The left branch at x<12 is (1)x+(-1); the right branch at x>12 is (-1)x+(23). Its explicit center value is f(12)=11. Is it continuous at 12?","answer":{"kind":"short_text","value":"Yes","explanation":"The side limits are [11, 11], and the center value is 11. All three must agree, giving Yes; agreement of the sides alone is insufficient."},"skills":["Continuity with Two Sides and a Defined Center Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0051","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"afd16758184475a2472ab3e694f449db938cf9d094d4cffb53259029e57ee553","family_id":"qf-continuity-removable-extension","seed":202610083051,"semantic_template":"qf-continuity-removable-extension/{\"inputs\":[\"-12\"],\"operation\":\"removable_continuous_extension\"}"},"fields":{"title":"Center Value for a Continuous Extension","stem":"The original quotient (x²−(-12)²)/(x−(-12)) is defined only away from -12. Choose the center value of a new continuous extension to all R.","answer":{"kind":"short_text","value":"-24","explanation":"The punctured limit is 2a=-24, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified."},"skills":["Center Value for a Continuous Extension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0052","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e77244932303ba56918697885838b0d1df834a02a174c917fea9543c9d84735","family_id":"qf-continuity-removable-extension","seed":202610083052,"semantic_template":"qf-continuity-removable-extension/{\"inputs\":[\"-11\"],\"operation\":\"removable_continuous_extension\"}"},"fields":{"title":"Center Value for a Continuous Extension","stem":"The original quotient (x²−(-11)²)/(x−(-11)) is defined only away from -11. Choose the center value of a new continuous extension to all R.","answer":{"kind":"short_text","value":"-22","explanation":"The punctured limit is 2a=-22, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified."},"skills":["Center Value for a Continuous Extension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0053","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"118fc24b4fa62f9427e256980e3f27765fe1ac5c073beefd94be44a2949c76dd","family_id":"qf-continuity-removable-extension","seed":202610083053,"semantic_template":"qf-continuity-removable-extension/{\"inputs\":[\"-10\"],\"operation\":\"removable_continuous_extension\"}"},"fields":{"title":"Center Value for a Continuous Extension","stem":"The original quotient (x²−(-10)²)/(x−(-10)) is defined only away from -10. Choose the center value of a new continuous extension to all R.","answer":{"kind":"short_text","value":"-20","explanation":"The punctured limit is 2a=-20, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified."},"skills":["Center Value for a Continuous Extension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0054","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a82fffc619dd186a047a04759e142c1ff263ca2de3252c51f5b5074507db668","family_id":"qf-continuity-removable-extension","seed":202610083054,"semantic_template":"qf-continuity-removable-extension/{\"inputs\":[\"-9\"],\"operation\":\"removable_continuous_extension\"}"},"fields":{"title":"Center Value for a Continuous Extension","stem":"The original quotient (x²−(-9)²)/(x−(-9)) is defined only away from -9. Choose the center value of a new continuous extension to all R.","answer":{"kind":"short_text","value":"-18","explanation":"The punctured limit is 2a=-18, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified."},"skills":["Center Value for a Continuous Extension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0055","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"01a77a558e7143d706aa05583cb0bcd1c6e2c8c6339d25d4e9245ee896f83a02","family_id":"qf-continuity-removable-extension","seed":202610083055,"semantic_template":"qf-continuity-removable-extension/{\"inputs\":[\"-8\"],\"operation\":\"removable_continuous_extension\"}"},"fields":{"title":"Center Value for a Continuous Extension","stem":"The original quotient (x²−(-8)²)/(x−(-8)) is defined only away from -8. Choose the center value of a new continuous extension to all R.","answer":{"kind":"short_text","value":"-16","explanation":"The punctured limit is 2a=-16, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified."},"skills":["Center Value for a Continuous Extension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0056","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"30488ca9f0d763ac1eba1cbd61f445cd499522eb6bcd2a618d07d28dadae92ea","family_id":"qf-continuity-removable-extension","seed":202610083056,"semantic_template":"qf-continuity-removable-extension/{\"inputs\":[\"-7\"],\"operation\":\"removable_continuous_extension\"}"},"fields":{"title":"Center Value for a Continuous Extension","stem":"The original quotient (x²−(-7)²)/(x−(-7)) is defined only away from -7. Choose the center value of a new continuous extension to all R.","answer":{"kind":"short_text","value":"-14","explanation":"The punctured limit is 2a=-14, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified."},"skills":["Center Value for a Continuous Extension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0057","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f021beb6f5b7cc8473cc6bb8efc3bc5eeec1d73e888a4cecf1fefd035f7bb0f","family_id":"qf-continuity-removable-extension","seed":202610083057,"semantic_template":"qf-continuity-removable-extension/{\"inputs\":[\"-6\"],\"operation\":\"removable_continuous_extension\"}"},"fields":{"title":"Center Value for a Continuous Extension","stem":"The original quotient (x²−(-6)²)/(x−(-6)) is defined only away from -6. Choose the center value of a new continuous extension to all R.","answer":{"kind":"short_text","value":"-12","explanation":"The punctured limit is 2a=-12, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified."},"skills":["Center Value for a Continuous Extension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0058","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9f15f6f94b75a07cb9301aca97551106b11eeb5eee98d27b3c0db2501071a3a8","family_id":"qf-continuity-removable-extension","seed":202610083058,"semantic_template":"qf-continuity-removable-extension/{\"inputs\":[\"-5\"],\"operation\":\"removable_continuous_extension\"}"},"fields":{"title":"Center Value for a Continuous Extension","stem":"The original quotient (x²−(-5)²)/(x−(-5)) is defined only away from -5. Choose the center value of a new continuous extension to all R.","answer":{"kind":"short_text","value":"-10","explanation":"The punctured limit is 2a=-10, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified."},"skills":["Center Value for a Continuous Extension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0059","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fda2c0d6f4d13996cc7efbc48811b364efe04d2a5adfcd2bfa05e7796e49c70d","family_id":"qf-continuity-removable-extension","seed":202610083059,"semantic_template":"qf-continuity-removable-extension/{\"inputs\":[\"-4\"],\"operation\":\"removable_continuous_extension\"}"},"fields":{"title":"Center Value for a Continuous Extension","stem":"The original quotient (x²−(-4)²)/(x−(-4)) is defined only away from -4. Choose the center value of a new continuous extension to all R.","answer":{"kind":"short_text","value":"-8","explanation":"The punctured limit is 2a=-8, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified."},"skills":["Center Value for a Continuous Extension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0060","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6edf5972989f1f3f684210842c70eaef5b4ebc0b10198ac5f5415132e1e6959e","family_id":"qf-continuity-removable-extension","seed":202610083060,"semantic_template":"qf-continuity-removable-extension/{\"inputs\":[\"-3\"],\"operation\":\"removable_continuous_extension\"}"},"fields":{"title":"Center Value for a Continuous Extension","stem":"The original quotient (x²−(-3)²)/(x−(-3)) is defined only away from -3. Choose the center value of a new continuous extension to all R.","answer":{"kind":"short_text","value":"-6","explanation":"The punctured limit is 2a=-6, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified."},"skills":["Center Value for a Continuous Extension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0061","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c359714559ce60a35f435b1b67d52bf94c21770fa73618fc9ec15cdbcb69e73","family_id":"qf-continuity-removable-extension","seed":202610083061,"semantic_template":"qf-continuity-removable-extension/{\"inputs\":[\"-2\"],\"operation\":\"removable_continuous_extension\"}"},"fields":{"title":"Center Value for a Continuous Extension","stem":"The original quotient (x²−(-2)²)/(x−(-2)) is defined only away from -2. Choose the center value of a new continuous extension to all R.","answer":{"kind":"short_text","value":"-4","explanation":"The punctured limit is 2a=-4, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified."},"skills":["Center Value for a Continuous Extension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0062","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b58cb630dea0a19b8d0e7c5e6352a72b291da951f3ecd076ca530fa10947a88d","family_id":"qf-continuity-removable-extension","seed":202610083062,"semantic_template":"qf-continuity-removable-extension/{\"inputs\":[\"-1\"],\"operation\":\"removable_continuous_extension\"}"},"fields":{"title":"Center Value for a Continuous Extension","stem":"The original quotient (x²−(-1)²)/(x−(-1)) is defined only away from -1. Choose the center value of a new continuous extension to all R.","answer":{"kind":"short_text","value":"-2","explanation":"The punctured limit is 2a=-2, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified."},"skills":["Center Value for a Continuous Extension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0063","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5828ed5147a6df3b06705df1d0810a5f9c025ca7aa33097d05b963277fd34878","family_id":"qf-continuity-removable-extension","seed":202610083063,"semantic_template":"qf-continuity-removable-extension/{\"inputs\":[\"0\"],\"operation\":\"removable_continuous_extension\"}"},"fields":{"title":"Center Value for a Continuous Extension","stem":"The original quotient (x²−(0)²)/(x−(0)) is defined only away from 0. Choose the center value of a new continuous extension to all R.","answer":{"kind":"short_text","value":"0","explanation":"The punctured limit is 2a=0, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified."},"skills":["Center Value for a Continuous Extension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0064","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be4888f4ee40d7eba6935a0b4725481af2f36f2716d3c51a487446c7809d1533","family_id":"qf-continuity-removable-extension","seed":202610083064,"semantic_template":"qf-continuity-removable-extension/{\"inputs\":[\"1\"],\"operation\":\"removable_continuous_extension\"}"},"fields":{"title":"Center Value for a Continuous Extension","stem":"The original quotient (x²−(1)²)/(x−(1)) is defined only away from 1. Choose the center value of a new continuous extension to all R.","answer":{"kind":"short_text","value":"2","explanation":"The punctured limit is 2a=2, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified."},"skills":["Center Value for a Continuous Extension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0065","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6d0d256f021ed3c4be66124b63b6ff33d0777ea0531b4e894b2733b05c28b3d8","family_id":"qf-continuity-removable-extension","seed":202610083065,"semantic_template":"qf-continuity-removable-extension/{\"inputs\":[\"2\"],\"operation\":\"removable_continuous_extension\"}"},"fields":{"title":"Center Value for a Continuous Extension","stem":"The original quotient (x²−(2)²)/(x−(2)) is defined only away from 2. Choose the center value of a new continuous extension to all R.","answer":{"kind":"short_text","value":"4","explanation":"The punctured limit is 2a=4, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified."},"skills":["Center Value for a Continuous Extension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0066","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"05e1ebe4e52532196492e418ac949b45d598dd5fb5cb83e5ff8b705e1c283b06","family_id":"qf-continuity-removable-extension","seed":202610083066,"semantic_template":"qf-continuity-removable-extension/{\"inputs\":[\"3\"],\"operation\":\"removable_continuous_extension\"}"},"fields":{"title":"Center Value for a Continuous Extension","stem":"The original quotient (x²−(3)²)/(x−(3)) is defined only away from 3. Choose the center value of a new continuous extension to all R.","answer":{"kind":"short_text","value":"6","explanation":"The punctured limit is 2a=6, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified."},"skills":["Center Value for a Continuous Extension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0067","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"125e036fcb8f8c861e1518b6807e6a054ad09c099d01c4471e8718ff6c35d23e","family_id":"qf-continuity-removable-extension","seed":202610083067,"semantic_template":"qf-continuity-removable-extension/{\"inputs\":[\"4\"],\"operation\":\"removable_continuous_extension\"}"},"fields":{"title":"Center Value for a Continuous Extension","stem":"The original quotient (x²−(4)²)/(x−(4)) is defined only away from 4. Choose the center value of a new continuous extension to all R.","answer":{"kind":"short_text","value":"8","explanation":"The punctured limit is 2a=8, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified."},"skills":["Center Value for a Continuous Extension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0068","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ddd4d4ee9e1b4f525d5cb683982d19d1c42a5110cf616616e7d955b997aff998","family_id":"qf-continuity-removable-extension","seed":202610083068,"semantic_template":"qf-continuity-removable-extension/{\"inputs\":[\"5\"],\"operation\":\"removable_continuous_extension\"}"},"fields":{"title":"Center Value for a Continuous Extension","stem":"The original quotient (x²−(5)²)/(x−(5)) is defined only away from 5. Choose the center value of a new continuous extension to all R.","answer":{"kind":"short_text","value":"10","explanation":"The punctured limit is 2a=10, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified."},"skills":["Center Value for a Continuous Extension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0069","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bbdaa8ab56e45956e78ff1f09d476901663eb2045fb5c87b4d8dfd308b0bc61e","family_id":"qf-continuity-removable-extension","seed":202610083069,"semantic_template":"qf-continuity-removable-extension/{\"inputs\":[\"6\"],\"operation\":\"removable_continuous_extension\"}"},"fields":{"title":"Center Value for a Continuous Extension","stem":"The original quotient (x²−(6)²)/(x−(6)) is defined only away from 6. Choose the center value of a new continuous extension to all R.","answer":{"kind":"short_text","value":"12","explanation":"The punctured limit is 2a=12, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified."},"skills":["Center Value for a Continuous Extension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0070","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c5f279445f41448751838aef9e64f2212e6167da2584afe9e951d40514a50188","family_id":"qf-continuity-removable-extension","seed":202610083070,"semantic_template":"qf-continuity-removable-extension/{\"inputs\":[\"7\"],\"operation\":\"removable_continuous_extension\"}"},"fields":{"title":"Center Value for a Continuous Extension","stem":"The original quotient (x²−(7)²)/(x−(7)) is defined only away from 7. Choose the center value of a new continuous extension to all R.","answer":{"kind":"short_text","value":"14","explanation":"The punctured limit is 2a=14, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified."},"skills":["Center Value for a Continuous Extension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0071","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"72df44eca8b66b6a38055000985fe167db956f025807a6be6d657b879bf04b5f","family_id":"qf-continuity-removable-extension","seed":202610083071,"semantic_template":"qf-continuity-removable-extension/{\"inputs\":[\"8\"],\"operation\":\"removable_continuous_extension\"}"},"fields":{"title":"Center Value for a Continuous Extension","stem":"The original quotient (x²−(8)²)/(x−(8)) is defined only away from 8. Choose the center value of a new continuous extension to all R.","answer":{"kind":"short_text","value":"16","explanation":"The punctured limit is 2a=16, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified."},"skills":["Center Value for a Continuous Extension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0072","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb65c61649f2e30ef27b62e5e1ff885b977625a5ed8ce727adf1ebbf2e950896","family_id":"qf-continuity-removable-extension","seed":202610083072,"semantic_template":"qf-continuity-removable-extension/{\"inputs\":[\"9\"],\"operation\":\"removable_continuous_extension\"}"},"fields":{"title":"Center Value for a Continuous Extension","stem":"The original quotient (x²−(9)²)/(x−(9)) is defined only away from 9. Choose the center value of a new continuous extension to all R.","answer":{"kind":"short_text","value":"18","explanation":"The punctured limit is 2a=18, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified."},"skills":["Center Value for a Continuous Extension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0073","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ea94908971f13ac11864d2ef87a1ebd4e3ccc81f49393ee3ba9126ecf8896b19","family_id":"qf-continuity-removable-extension","seed":202610083073,"semantic_template":"qf-continuity-removable-extension/{\"inputs\":[\"10\"],\"operation\":\"removable_continuous_extension\"}"},"fields":{"title":"Center Value for a Continuous Extension","stem":"The original quotient (x²−(10)²)/(x−(10)) is defined only away from 10. Choose the center value of a new continuous extension to all R.","answer":{"kind":"short_text","value":"20","explanation":"The punctured limit is 2a=20, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified."},"skills":["Center Value for a Continuous Extension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0074","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4250ae77b4a0f67c169538283bc3518c648a930dbca80115603ea09501a20b5b","family_id":"qf-continuity-removable-extension","seed":202610083074,"semantic_template":"qf-continuity-removable-extension/{\"inputs\":[\"11\"],\"operation\":\"removable_continuous_extension\"}"},"fields":{"title":"Center Value for a Continuous Extension","stem":"The original quotient (x²−(11)²)/(x−(11)) is defined only away from 11. Choose the center value of a new continuous extension to all R.","answer":{"kind":"short_text","value":"22","explanation":"The punctured limit is 2a=22, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified."},"skills":["Center Value for a Continuous Extension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0075","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ab335e51880be5de72a157a677fedebe9f3c131110157313eb78e7056eb08307","family_id":"qf-continuity-removable-extension","seed":202610083075,"semantic_template":"qf-continuity-removable-extension/{\"inputs\":[\"12\"],\"operation\":\"removable_continuous_extension\"}"},"fields":{"title":"Center Value for a Continuous Extension","stem":"The original quotient (x²−(12)²)/(x−(12)) is defined only away from 12. Choose the center value of a new continuous extension to all R.","answer":{"kind":"short_text","value":"24","explanation":"The punctured limit is 2a=24, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified."},"skills":["Center Value for a Continuous Extension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0076","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b8c4a351b2fea22f13807ac97d9218db6fa40a8cc22eef3728fbe8daa9281bef","family_id":"qf-continuity-point-defects","seed":202610083076,"semantic_template":"qf-continuity-point-defects/{\"inputs\":[\"0\",\"0\"],\"operation\":\"point_defects\"}"},"fields":{"title":"One-Sided Limit Defects Relative to the Point Value","stem":"On all R, The left branch at x<-12 is (1)x+(0); the right branch at x>-12 is (-1)x+(-24). The center value is -12. Give (left limit−center value, right limit−center value).","answer":{"kind":"expression","value":"(0, 0)","explanation":"The side limits are -12 and -12; subtract the independently given center value to obtain (0, 0). Both defects must be zero for continuity in this model."},"skills":["One-Sided Limit Defects Relative to the Point Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0077","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"37e8035d61365b1e239da0bf066133ebc7a604aadac3a742e1d53aa6ffea34ce","family_id":"qf-continuity-point-defects","seed":202610083077,"semantic_template":"qf-continuity-point-defects/{\"inputs\":[\"1\",\"-1\"],\"operation\":\"point_defects\"}"},"fields":{"title":"One-Sided Limit Defects Relative to the Point Value","stem":"On all R, The left branch at x<-11 is (2)x+(12); the right branch at x>-11 is (-2)x+(-34). The center value is -11. Give (left limit−center value, right limit−center value).","answer":{"kind":"expression","value":"(1, -1)","explanation":"The side limits are -10 and -12; subtract the independently given center value to obtain (1, -1). Both defects must be zero for continuity in this model."},"skills":["One-Sided Limit Defects Relative to the Point Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0078","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5ec79b91101a837d435ee13a532662eeb38ce2e45f7403e7f1a18c000dcf9368","family_id":"qf-continuity-point-defects","seed":202610083078,"semantic_template":"qf-continuity-point-defects/{\"inputs\":[\"2\",\"3\"],\"operation\":\"point_defects\"}"},"fields":{"title":"One-Sided Limit Defects Relative to the Point Value","stem":"On all R, The left branch at x<-10 is (3)x+(22); the right branch at x>-10 is (-3)x+(-37). The center value is -10. Give (left limit−center value, right limit−center value).","answer":{"kind":"expression","value":"(2, 3)","explanation":"The side limits are -8 and -7; subtract the independently given center value to obtain (2, 3). Both defects must be zero for continuity in this model."},"skills":["One-Sided Limit Defects Relative to the Point Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0079","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"39d33c37fa8e1975b9586a84f116f1c805d98810fb7ee1e4f66cee6131906d10","family_id":"qf-continuity-point-defects","seed":202610083079,"semantic_template":"qf-continuity-point-defects/{\"inputs\":[\"3\",\"-3\"],\"operation\":\"point_defects\"}"},"fields":{"title":"One-Sided Limit Defects Relative to the Point Value","stem":"On all R, The left branch at x<-9 is (4)x+(30); the right branch at x>-9 is (-1)x+(-21). The center value is -9. Give (left limit−center value, right limit−center value).","answer":{"kind":"expression","value":"(3, -3)","explanation":"The side limits are -6 and -12; subtract the independently given center value to obtain (3, -3). Both defects must be zero for continuity in this model."},"skills":["One-Sided Limit Defects Relative to the Point Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0080","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e7a2d25471434405f87a08d9e6f51dc4753b42cffe0ba6968432ae294dc1dcd1","family_id":"qf-continuity-point-defects","seed":202610083080,"semantic_template":"qf-continuity-point-defects/{\"inputs\":[\"4\",\"5\"],\"operation\":\"point_defects\"}"},"fields":{"title":"One-Sided Limit Defects Relative to the Point Value","stem":"On all R, The left branch at x<-8 is (1)x+(4); the right branch at x>-8 is (-2)x+(-19). The center value is -8. Give (left limit−center value, right limit−center value).","answer":{"kind":"expression","value":"(4, 5)","explanation":"The side limits are -4 and -3; subtract the independently given center value to obtain (4, 5). Both defects must be zero for continuity in this model."},"skills":["One-Sided Limit Defects Relative to the Point Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0081","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"397a5d1dc4736ec348d44943fdae056d8ed759bdb901f6cea6d7ae6f28e3fd0d","family_id":"qf-continuity-point-defects","seed":202610083081,"semantic_template":"qf-continuity-point-defects/{\"inputs\":[\"5\",\"-5\"],\"operation\":\"point_defects\"}"},"fields":{"title":"One-Sided Limit Defects Relative to the Point Value","stem":"On all R, The left branch at x<-7 is (2)x+(12); the right branch at x>-7 is (-3)x+(-33). The center value is -7. Give (left limit−center value, right limit−center value).","answer":{"kind":"expression","value":"(5, -5)","explanation":"The side limits are -2 and -12; subtract the independently given center value to obtain (5, -5). Both defects must be zero for continuity in this model."},"skills":["One-Sided Limit Defects Relative to the Point Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0082","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec244698745e119772b363bc4535cd4decb80ec5a00334a8ad61f58124f76bab","family_id":"qf-continuity-point-defects","seed":202610083082,"semantic_template":"qf-continuity-point-defects/{\"inputs\":[\"6\",\"7\"],\"operation\":\"point_defects\"}"},"fields":{"title":"One-Sided Limit Defects Relative to the Point Value","stem":"On all R, The left branch at x<-6 is (3)x+(18); the right branch at x>-6 is (-1)x+(-5). The center value is -6. Give (left limit−center value, right limit−center value).","answer":{"kind":"expression","value":"(6, 7)","explanation":"The side limits are 0 and 1; subtract the independently given center value to obtain (6, 7). Both defects must be zero for continuity in this model."},"skills":["One-Sided Limit Defects Relative to the Point Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0083","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a7192b9b0c1edf4006b5cdf83db756814948f0cb8bd117fe73e9a5c46cbc861e","family_id":"qf-continuity-point-defects","seed":202610083083,"semantic_template":"qf-continuity-point-defects/{\"inputs\":[\"7\",\"-7\"],\"operation\":\"point_defects\"}"},"fields":{"title":"One-Sided Limit Defects Relative to the Point Value","stem":"On all R, The left branch at x<-5 is (4)x+(22); the right branch at x>-5 is (-2)x+(-22). The center value is -5. Give (left limit−center value, right limit−center value).","answer":{"kind":"expression","value":"(7, -7)","explanation":"The side limits are 2 and -12; subtract the independently given center value to obtain (7, -7). Both defects must be zero for continuity in this model."},"skills":["One-Sided Limit Defects Relative to the Point Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0084","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"07d8a5233debc3919a38b8d514f298a097dffb90f69c32285f9f976f79fb0b31","family_id":"qf-continuity-point-defects","seed":202610083084,"semantic_template":"qf-continuity-point-defects/{\"inputs\":[\"8\",\"9\"],\"operation\":\"point_defects\"}"},"fields":{"title":"One-Sided Limit Defects Relative to the Point Value","stem":"On all R, The left branch at x<-4 is (1)x+(8); the right branch at x>-4 is (-3)x+(-7). The center value is -4. Give (left limit−center value, right limit−center value).","answer":{"kind":"expression","value":"(8, 9)","explanation":"The side limits are 4 and 5; subtract the independently given center value to obtain (8, 9). Both defects must be zero for continuity in this model."},"skills":["One-Sided Limit Defects Relative to the Point Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0085","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e50bf9a09aaa8d1209a7072c2eb13460194604bc21d2e9bfefec91851777572","family_id":"qf-continuity-point-defects","seed":202610083085,"semantic_template":"qf-continuity-point-defects/{\"inputs\":[\"9\",\"-9\"],\"operation\":\"point_defects\"}"},"fields":{"title":"One-Sided Limit Defects Relative to the Point Value","stem":"On all R, The left branch at x<-3 is (2)x+(12); the right branch at x>-3 is (-1)x+(-15). The center value is -3. Give (left limit−center value, right limit−center value).","answer":{"kind":"expression","value":"(9, -9)","explanation":"The side limits are 6 and -12; subtract the independently given center value to obtain (9, -9). Both defects must be zero for continuity in this model."},"skills":["One-Sided Limit Defects Relative to the Point Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0086","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8064e3126f5447b1db6770d570f52a93a5e35cb70c006bc0a43520802cfe10ae","family_id":"qf-continuity-point-defects","seed":202610083086,"semantic_template":"qf-continuity-point-defects/{\"inputs\":[\"10\",\"11\"],\"operation\":\"point_defects\"}"},"fields":{"title":"One-Sided Limit Defects Relative to the Point Value","stem":"On all R, The left branch at x<-2 is (3)x+(14); the right branch at x>-2 is (-2)x+(5). The center value is -2. Give (left limit−center value, right limit−center value).","answer":{"kind":"expression","value":"(10, 11)","explanation":"The side limits are 8 and 9; subtract the independently given center value to obtain (10, 11). Both defects must be zero for continuity in this model."},"skills":["One-Sided Limit Defects Relative to the Point Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0087","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"14eb5391ae357af3010c4d40fdb5cffdca5ac152ef959c7ee2731434696184a9","family_id":"qf-continuity-point-defects","seed":202610083087,"semantic_template":"qf-continuity-point-defects/{\"inputs\":[\"11\",\"-11\"],\"operation\":\"point_defects\"}"},"fields":{"title":"One-Sided Limit Defects Relative to the Point Value","stem":"On all R, The left branch at x<-1 is (4)x+(14); the right branch at x>-1 is (-3)x+(-15). The center value is -1. Give (left limit−center value, right limit−center value).","answer":{"kind":"expression","value":"(11, -11)","explanation":"The side limits are 10 and -12; subtract the independently given center value to obtain (11, -11). Both defects must be zero for continuity in this model."},"skills":["One-Sided Limit Defects Relative to the Point Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0088","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a3832aeb56edb0f95883f118e9140ee9a0ff7b5fbe4617fef986cb9846f35a56","family_id":"qf-continuity-point-defects","seed":202610083088,"semantic_template":"qf-continuity-point-defects/{\"inputs\":[\"12\",\"13\"],\"operation\":\"point_defects\"}"},"fields":{"title":"One-Sided Limit Defects Relative to the Point Value","stem":"On all R, The left branch at x<0 is (1)x+(12); the right branch at x>0 is (-1)x+(13). The center value is 0. Give (left limit−center value, right limit−center value).","answer":{"kind":"expression","value":"(12, 13)","explanation":"The side limits are 12 and 13; subtract the independently given center value to obtain (12, 13). Both defects must be zero for continuity in this model."},"skills":["One-Sided Limit Defects Relative to the Point Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0089","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"61b9dbfe8b9a1ccf89012d5b3488ccb06e741798951a3f40f385bb70a2d76c3f","family_id":"qf-continuity-point-defects","seed":202610083089,"semantic_template":"qf-continuity-point-defects/{\"inputs\":[\"13\",\"-13\"],\"operation\":\"point_defects\"}"},"fields":{"title":"One-Sided Limit Defects Relative to the Point Value","stem":"On all R, The left branch at x<1 is (2)x+(12); the right branch at x>1 is (-2)x+(-10). The center value is 1. Give (left limit−center value, right limit−center value).","answer":{"kind":"expression","value":"(13, -13)","explanation":"The side limits are 14 and -12; subtract the independently given center value to obtain (13, -13). Both defects must be zero for continuity in this model."},"skills":["One-Sided Limit Defects Relative to the Point Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0090","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aca1fec84183bf297c98397e0f10b130ade9e77ce2474e7cb7dce5b8c3ad1235","family_id":"qf-continuity-point-defects","seed":202610083090,"semantic_template":"qf-continuity-point-defects/{\"inputs\":[\"14\",\"15\"],\"operation\":\"point_defects\"}"},"fields":{"title":"One-Sided Limit Defects Relative to the Point Value","stem":"On all R, The left branch at x<2 is (3)x+(10); the right branch at x>2 is (-3)x+(23). The center value is 2. Give (left limit−center value, right limit−center value).","answer":{"kind":"expression","value":"(14, 15)","explanation":"The side limits are 16 and 17; subtract the independently given center value to obtain (14, 15). Both defects must be zero for continuity in this model."},"skills":["One-Sided Limit Defects Relative to the Point Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0091","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba2ffe17ae87298bb113e211e97c9765734bd81c50b45474e2ef31e82e38b5ad","family_id":"qf-continuity-point-defects","seed":202610083091,"semantic_template":"qf-continuity-point-defects/{\"inputs\":[\"15\",\"-15\"],\"operation\":\"point_defects\"}"},"fields":{"title":"One-Sided Limit Defects Relative to the Point Value","stem":"On all R, The left branch at x<3 is (4)x+(6); the right branch at x>3 is (-1)x+(-9). The center value is 3. Give (left limit−center value, right limit−center value).","answer":{"kind":"expression","value":"(15, -15)","explanation":"The side limits are 18 and -12; subtract the independently given center value to obtain (15, -15). Both defects must be zero for continuity in this model."},"skills":["One-Sided Limit Defects Relative to the Point Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0092","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c495e7705835e7744269a1cbfb02591550be245910cd1b6e9b0d9d01a70e1c21","family_id":"qf-continuity-point-defects","seed":202610083092,"semantic_template":"qf-continuity-point-defects/{\"inputs\":[\"16\",\"17\"],\"operation\":\"point_defects\"}"},"fields":{"title":"One-Sided Limit Defects Relative to the Point Value","stem":"On all R, The left branch at x<4 is (1)x+(16); the right branch at x>4 is (-2)x+(29). The center value is 4. Give (left limit−center value, right limit−center value).","answer":{"kind":"expression","value":"(16, 17)","explanation":"The side limits are 20 and 21; subtract the independently given center value to obtain (16, 17). Both defects must be zero for continuity in this model."},"skills":["One-Sided Limit Defects Relative to the Point Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0093","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"19dafb94a7493c2c0c4a7a7a9c5abed5e2e68abe39642c22667b3c5e26e69492","family_id":"qf-continuity-point-defects","seed":202610083093,"semantic_template":"qf-continuity-point-defects/{\"inputs\":[\"17\",\"-17\"],\"operation\":\"point_defects\"}"},"fields":{"title":"One-Sided Limit Defects Relative to the Point Value","stem":"On all R, The left branch at x<5 is (2)x+(12); the right branch at x>5 is (-3)x+(3). The center value is 5. Give (left limit−center value, right limit−center value).","answer":{"kind":"expression","value":"(17, -17)","explanation":"The side limits are 22 and -12; subtract the independently given center value to obtain (17, -17). Both defects must be zero for continuity in this model."},"skills":["One-Sided Limit Defects Relative to the Point Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0094","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b8d33eab5c6f5ba25f6fe699de8c9ba184c6de111cd46941038a07063fc1c65","family_id":"qf-continuity-point-defects","seed":202610083094,"semantic_template":"qf-continuity-point-defects/{\"inputs\":[\"18\",\"19\"],\"operation\":\"point_defects\"}"},"fields":{"title":"One-Sided Limit Defects Relative to the Point Value","stem":"On all R, The left branch at x<6 is (3)x+(6); the right branch at x>6 is (-1)x+(31). The center value is 6. Give (left limit−center value, right limit−center value).","answer":{"kind":"expression","value":"(18, 19)","explanation":"The side limits are 24 and 25; subtract the independently given center value to obtain (18, 19). Both defects must be zero for continuity in this model."},"skills":["One-Sided Limit Defects Relative to the Point Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0095","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bd8cadd60f9693e731d0ada225ebfd8a8f924ff53b3cbfa3afc82f6b7e372024","family_id":"qf-continuity-point-defects","seed":202610083095,"semantic_template":"qf-continuity-point-defects/{\"inputs\":[\"19\",\"-19\"],\"operation\":\"point_defects\"}"},"fields":{"title":"One-Sided Limit Defects Relative to the Point Value","stem":"On all R, The left branch at x<7 is (4)x+(-2); the right branch at x>7 is (-2)x+(2). The center value is 7. Give (left limit−center value, right limit−center value).","answer":{"kind":"expression","value":"(19, -19)","explanation":"The side limits are 26 and -12; subtract the independently given center value to obtain (19, -19). Both defects must be zero for continuity in this model."},"skills":["One-Sided Limit Defects Relative to the Point Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0096","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"44d80c21b99157d5c7a19fad904a4231b5cbffeed13e313462e629d927551ebd","family_id":"qf-continuity-point-defects","seed":202610083096,"semantic_template":"qf-continuity-point-defects/{\"inputs\":[\"20\",\"21\"],\"operation\":\"point_defects\"}"},"fields":{"title":"One-Sided Limit Defects Relative to the Point Value","stem":"On all R, The left branch at x<8 is (1)x+(20); the right branch at x>8 is (-3)x+(53). The center value is 8. Give (left limit−center value, right limit−center value).","answer":{"kind":"expression","value":"(20, 21)","explanation":"The side limits are 28 and 29; subtract the independently given center value to obtain (20, 21). Both defects must be zero for continuity in this model."},"skills":["One-Sided Limit Defects Relative to the Point Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0097","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c6b7eea9f6986a244694fcd4221e889b376df2b9ee00bae2683452a855d1da8","family_id":"qf-continuity-point-defects","seed":202610083097,"semantic_template":"qf-continuity-point-defects/{\"inputs\":[\"21\",\"-21\"],\"operation\":\"point_defects\"}"},"fields":{"title":"One-Sided Limit Defects Relative to the Point Value","stem":"On all R, The left branch at x<9 is (2)x+(12); the right branch at x>9 is (-1)x+(-3). The center value is 9. Give (left limit−center value, right limit−center value).","answer":{"kind":"expression","value":"(21, -21)","explanation":"The side limits are 30 and -12; subtract the independently given center value to obtain (21, -21). Both defects must be zero for continuity in this model."},"skills":["One-Sided Limit Defects Relative to the Point Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0098","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e19f6243ec8a6e90b5d25a8909acb5a5cc10e3d4ad6d08cea7bd660a6b803124","family_id":"qf-continuity-point-defects","seed":202610083098,"semantic_template":"qf-continuity-point-defects/{\"inputs\":[\"22\",\"23\"],\"operation\":\"point_defects\"}"},"fields":{"title":"One-Sided Limit Defects Relative to the Point Value","stem":"On all R, The left branch at x<10 is (3)x+(2); the right branch at x>10 is (-2)x+(53). The center value is 10. Give (left limit−center value, right limit−center value).","answer":{"kind":"expression","value":"(22, 23)","explanation":"The side limits are 32 and 33; subtract the independently given center value to obtain (22, 23). Both defects must be zero for continuity in this model."},"skills":["One-Sided Limit Defects Relative to the Point Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0099","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2827968aa5813482c3d0d5c6c8e5bae4ef169c7b174844795c0fa0bcfcf108f8","family_id":"qf-continuity-point-defects","seed":202610083099,"semantic_template":"qf-continuity-point-defects/{\"inputs\":[\"23\",\"-23\"],\"operation\":\"point_defects\"}"},"fields":{"title":"One-Sided Limit Defects Relative to the Point Value","stem":"On all R, The left branch at x<11 is (4)x+(-10); the right branch at x>11 is (-3)x+(21). The center value is 11. Give (left limit−center value, right limit−center value).","answer":{"kind":"expression","value":"(23, -23)","explanation":"The side limits are 34 and -12; subtract the independently given center value to obtain (23, -23). Both defects must be zero for continuity in this model."},"skills":["One-Sided Limit Defects Relative to the Point Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r03-continuity-0100","batch_id":"batch-20261007-advance-20261008-r03-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b70d5ff985a191a5a5d732edecf244a8f57d0a0668f5c1dfdb04c01267162be4","family_id":"qf-continuity-point-defects","seed":202610083100,"semantic_template":"qf-continuity-point-defects/{\"inputs\":[\"24\",\"25\"],\"operation\":\"point_defects\"}"},"fields":{"title":"One-Sided Limit Defects Relative to the Point Value","stem":"On all R, The left branch at x<12 is (1)x+(24); the right branch at x>12 is (-1)x+(49). The center value is 12. Give (left limit−center value, right limit−center value).","answer":{"kind":"expression","value":"(24, 25)","explanation":"The side limits are 36 and 37; subtract the independently given center value to obtain (24, 25). Both defects must be zero for continuity in this model."},"skills":["One-Sided Limit Defects Relative to the Point Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}