{"schema_version":"ocme-site-results-v1","locale":"en","page_type":"derivations","facts":{"questions":36000,"reviewed":36000,"display_candidates":0,"knowledge":302,"guides":299,"paths":59,"batches":360,"families":1441,"atlas":302,"core":80,"evidence":9,"formal_evidence":5,"formal_subjects":4,"lean_files":6,"lean_toolchain":"leanprover/lean4:v4.30.0","question_state":"candidate","english_state":"ai_candidate","question_Lean":"NOT_RUN","canonical_truth_promotion":false},"filters":{"q":"mko-continuity"},"kind":"guides","total":1,"pages":1,"page":1,"entries":[{"id":"derivation-mko-continuity-01","title":{"zh-Hant":"連續：分支函數的中心值與可去商的唯一補點","en":"Continuity: centre values of branch functions and the unique continuous extension across a removable singularity"},"summary":{"zh-Hant":"(1) 對 f_V(x)=2x+1（x<1）、f_V(1)=V、f_V(x)=5x−2（x>1），證明 V=3 時 f_V 在 1 連續，V=4 時不連續；(2) 對 g(x)=2x+1（x<1）、g(1)=3、g(x)=5x−1（x>1），證明 g 在 1 不連續；(3) 對 q(x)=(x²−4)/(x−2)（論域 ℝ∖{2}），證明全實數連續延拓在 2 的值必為 4，且 4 確實可行。","en":"(1) For f_V(x)=2x+1 (x<1), f_V(1)=V, f_V(x)=5x−2 (x>1), prove that f_V is continuous at 1 when V=3 and not when V=4; (2) for g(x)=2x+1 (x<1), g(1)=3, g(x)=5x−1 (x>1), prove that g is not continuous at 1; (3) for q(x)=(x²−4)/(x−2) on ℝ∖{2}, prove that a continuous extension to all real numbers must take the value 4 at 2, and that 4 does work."},"mko_ids":["mko-continuity"]}],"title":"Concept derivation guides"}