{ "schema_version": "ocme-derivation-reader-v1", "guide_id": "derivation-mko-uniform-limit-continuity-theorem-01", "mko_id": "mko-uniform-limit-continuity-theorem", "title": { "zh-Hant": "固定一個近似函數的 epsilon/3 傳遞", "en": "An epsilon/3 transfer through one fixed approximant" }, "contract": { "schema_version": "ocme-derivation-guide-v0.1", "guide_id": "derivation-mko-uniform-limit-continuity-theorem-01", "source": { "mko_id": "mko-uniform-limit-continuity-theorem", "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f", "sha256": "7dd17546858c30c475f0f2eead9893cb35e9e8ed4969e65933c7a8fd7a621632", "english_sha256": "6c9744f93c7afdccbc56b487121dbdd22b7d603cb95927b0a7147b9456c1cd5d" }, "mode": "theorem_proof", "audience_prerequisites": [ "algebra", "polynomial", "epsilon", "continuity_def" ], "goal": { "zh-Hant": "對 [0,1] 上的仿射擾動證一致收斂，並逐步傳遞連續性。", "en": "Prove uniform convergence of affine perturbations on [0,1] and transfer continuity step by step." }, "givens": [ { "id": "G1", "statement": { "zh-Hant": "f_n(x)=2x+1+(3x-1)/(n+1)，n≥0，x∈[0,1]；f(x)=2x+1。", "en": "f_n(x)=2*x+1+(3*x-1)/(n+1), n≥0, x in [0,1]; f(x)=2*x+1." } } ], "steps": [ { "id": "S01", "uses": [ "G1" ], "before_tex": "f_n(x)=(2+3/(n+1))x+1-1/(n+1)", "after_tex": "f_n\\text{ continuous on }[0,1]\\quad\\forall n\\ge0", "relation": "implies", "rule": "polynomial", "conditions": [], "explanation": { "zh-Hant": "每個 n 分母正且是仿射多項式，逐個函數連續性成立。", "en": "Each denominator is positive and each function affine, proving continuity of every approximant." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1" ], "explanation": { "zh-Hant": "一致收斂本身不能替代這項前提。", "en": "Uniform convergence alone cannot replace this premise." } }, "leaf_basis": "polynomial" }, { "id": "S02", "uses": [ "G1" ], "before_tex": "0\\le x\\le1", "after_tex": "-1\\le3x-1\\le2,\\quad\\sup_{[0,1]}|f_n-f|=2/(n+1)", "relation": "implies", "rule": "algebra", "conditions": [], "explanation": { "zh-Hant": "絕對值至多 2 且 x=1 取到，故 sup 是精確等式。", "en": "The magnitude is at most two and attained at x=1, giving the exact supremum." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1" ], "explanation": { "zh-Hant": "不把有限網格最大值當完整區間 sup。", "en": "No finite-grid maximum is substituted for the whole-interval supremum." } }, "leaf_basis": "algebra" }, { "id": "S03", "uses": [ "S02" ], "before_tex": "\\varepsilon>0,\\quad N=\\lfloor6/\\varepsilon\\rfloor+1", "after_tex": "\\sup_{x\\in[0,1]}|f_n(x)-f(x)|<\\varepsilon/3\\quad(n\\ge N)", "relation": "implies", "rule": "epsilon", "conditions": [], "explanation": { "zh-Hant": "同一 N 只依 epsilon，控制所有點與所有後項。", "en": "One N depending only on epsilon controls all points and all later terms." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "S02" ], "explanation": { "zh-Hant": "不是先固定 x 再選 N。", "en": "N is not chosen after fixing x." } }, "leaf_basis": "epsilon" }, { "id": "S04", "uses": [ "G1", "S03" ], "before_tex": "x_0\\in[0,1],\\quad L_N=2+3/(N+1)>0", "after_tex": "\\delta=\\varepsilon/(3L_N)>0", "relation": "implies", "rule": "algebra", "conditions": [], "explanation": { "zh-Hant": "任取欲證連續的 x_0，固定前步選好的單一近似函數 f_N。", "en": "Choose an arbitrary point x_0 in the domain at which continuity is to be proved, and fix the single approximant f_N selected above." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "S03" ], "explanation": { "zh-Hant": "N 已固定才取 delta，delta 可依 N 與 epsilon。", "en": "N is fixed before choosing delta, which may depend on N and epsilon." } }, "leaf_basis": "algebra" }, { "id": "S05", "uses": [ "S01", "S04" ], "before_tex": "x\\in[0,1],\\quad|x-x_0|<\\delta", "after_tex": "|f_N(x)-f_N(x_0)|=L_N|x-x_0|<\\varepsilon/3", "relation": "implies", "rule": "algebra", "conditions": [], "explanation": { "zh-Hant": "仿射差值給固定近似函數的精確連續界。", "en": "The affine difference supplies the exact continuity bound for the fixed approximant." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "S01", "S04" ], "explanation": { "zh-Hant": "端點也使用相對域中的 x，不需越出 [0,1]。", "en": "At endpoints, x remains in the relative domain [0,1]." } }, "leaf_basis": "algebra" }, { "id": "S06", "uses": [ "S03", "S05" ], "before_tex": "|f(x)-f(x_0)|\\le|f(x)-f_N(x)|+|f_N(x)-f_N(x_0)|+|f_N(x_0)-f(x_0)|", "after_tex": "|f(x)-f(x_0)|<\\varepsilon/3+\\varepsilon/3+\\varepsilon/3=\\varepsilon", "relation": "implies", "rule": "algebra", "conditions": [], "explanation": { "zh-Hant": "兩個一致誤差加一個固定函數連續誤差，三角不等式完成傳遞。", "en": "Two uniform errors and one fixed-function continuity error combine by the triangle inequality." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "S03", "S05" ], "explanation": { "zh-Hant": "兩端的近似誤差都需要同一 N 對全部 x 有效。", "en": "Both approximation errors require the same N valid for every x." } }, "leaf_basis": "algebra" }, { "id": "S07", "uses": [ "S04", "S06", "continuity_def" ], "before_tex": "x_0\\in[0,1]\\text{ arbitrary},\\quad\\forall\\varepsilon>0\\ \\exists\\delta>0", "after_tex": "f\\text{ continuous on }[0,1]", "relation": "definition", "rule": "continuity_def", "conditions": [], "explanation": { "zh-Hant": "x_0 任意，對每點完成 epsilon-delta 定義；一般同樣傳遞只需每項連續及一致誤差。", "en": "Arbitrariness of x_0 establishes continuity everywhere; the same transfer in general only needs approximant continuity and uniform error." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "S04", "S06", "continuity_def" ], "explanation": { "zh-Hant": "未要求所有 f_n 共用一個 delta。", "en": "No common delta for all f_n is required." } }, "leaf_basis": "continuity_def" }, { "id": "S08", "uses": [ "G1" ], "before_tex": "u_n(x)=x^n\\ (n\\ge1),\\quad u(x)=0\\ (x<1),\\ u(1)=1", "after_tex": "u_n\\to u\\text{ pointwise but }|u_n(2^{-1/n})-u(2^{-1/n})|=1/2", "relation": "implies", "rule": "algebra", "conditions": [], "explanation": { "zh-Hant": "連續冪函數逐點極限在 1 不連續；每個 n 仍有誤差 1/2 的域內點，故不一致。", "en": "Continuous powers have a pointwise limit discontinuous at one; every n has a domain point with error 1/2, so convergence is not uniform." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1" ], "explanation": { "zh-Hant": "點可依 n 以否定 sup 控制，並非只核一個固定點。", "en": "The point may depend on n when refuting supremum control, unlike a fixed-point test." } }, "leaf_basis": "algebra" } ], "result": { "zh-Hant": "精確 sup=2/(n+1) 證一致收斂；epsilon/3 傳遞證極限連續，沒有共用 delta 的要求。", "en": "The exact supremum 2/(n+1) proves uniform convergence; epsilon/3 transfer proves continuity of the limit without a shared-delta requirement." }, "negative_controls": [ { "claim": { "zh-Hant": "逐點極限總是連續。", "en": "Pointwise limits are always continuous." }, "expected": "reject", "reason": { "zh-Hant": "x^n 的極限在 1 不連續。", "en": "The limit of x^n is discontinuous at one." } }, { "claim": { "zh-Hant": "一致收斂不需各項連續也保極限連續。", "en": "Uniform convergence preserves continuity without continuous terms." }, "expected": "reject", "reason": { "zh-Hant": "同一不連續函數的常值函數序列誤差恆零，極限仍不連續。", "en": "A sequence constantly equal to one discontinuous function has zero uniform error and the same discontinuous limit." } }, { "claim": { "zh-Hant": "要所有 f_n 用同一 delta 才能證定理。", "en": "All f_n need the same delta for the theorem." }, "expected": "reject", "reason": { "zh-Hant": "S04-S06 只固定一個 f_N 已足夠。", "en": "S04-S06 only fix one f_N, which suffices." } } ], "status": "candidate", "gaps": [], "verification": { "structure": "checked_local_script", "mathematics": "candidate_argument", "formalization": "not_formalized" }, "rule_glossary": [ { "id": "mko_definition", "kind": "cited", "statement": { "zh-Hant": "在共同domain上，每個f_n連續且sup_x |f_n(x)-f(x)|→0。題族用[0,1]上的affine perturbation、exact sup error與epsilon-N證書重算，並以epsilon/3 transfer把一個固定f_N的continuity傳給f。pointwise-only的x^n、uniform收斂到discontinuous limit但各項不連續、以及有限sample closeness都明示為失敗邊界。", "en": "Every approximating f_n is continuous on the same declared domain and sup_x|f_n(x)−f(x)| tends to 0. Exact families use affine perturbations on[0,1], exact supremum errors and epsilon–N certificates. An epsilon/3 transfer fixes one continuous approximant f_N and passes its continuity to f. Pointwise-only powers, uniform sequences whose terms are discontinuous and finite-grid agreement do not supply both theorem hypotheses." }, "source": { "mko_id": "mko-uniform-limit-continuity-theorem", "sha256": "7dd17546858c30c475f0f2eead9893cb35e9e8ed4969e65933c7a8fd7a621632", "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f" } }, { "id": "algebra", "kind": "prerequisite", "statement": { "zh-Hant": "實數域的等式運算；除法前分母必非零，不等式乘正數保持方向。", "en": "Use real-field identities; division requires a nonzero denominator and multiplication by a positive number preserves inequalities." } }, { "id": "polynomial", "kind": "prerequisite", "statement": { "zh-Hant": "多項式在實數上連續且可逐項微分；d(x^k)/dx=k x^(k-1)，常數導數為零。", "en": "Real polynomials are continuous and differentiate termwise: the derivative of x^k is k*x^(k-1), and constants differentiate to zero." } }, { "id": "epsilon", "kind": "prerequisite", "statement": { "zh-Hant": "收斂表示每個正 epsilon 有一個整數 N，使全部 n≥N 的誤差嚴格小於 epsilon；使用實數 Archimedean 性。", "en": "Convergence requires for every positive epsilon an integer N controlling every n≥N by a strict error bound; use the Archimedean property of the reals." } }, { "id": "continuity_def", "kind": "prerequisite", "statement": { "zh-Hant": "在共同域，f 於 x_0 連續指每個正 epsilon 有正 delta 控制全部域內 |x-x_0|