第 27014 題：一致極限保持連續

原題與已知條件

共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-continuity-transfer；資料={"n":14,"lipschitz":4,"error_bound":[1,17],"epsilon":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — 共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-continuity-transfer；資料={"n":14,"lipschitz":4,"error_bound":[1,17],"epsilon":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

定義與適用範圍

沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。

• mko-uniform-limit-continuity-theorem

逐步推導

source roles = [uniform-continuity-transfer]

derived = [[N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, certificate]]

Uniform ORIGINAL three times uniform error := mul(3, 1/17)

Uniform ORIGINAL three times uniform error = 3/17

Uniform ORIGINAL three times Lipschitz constant := mul(3, 4)

Uniform ORIGINAL three times Lipschitz constant = 12

Uniform ORIGINAL epsilon over threeL := div(1, 12)

Uniform ORIGINAL epsilon over threeL = 1/12

Uniform ORIGINAL twice uniform error := mul(2, 1/17)

Uniform ORIGINAL twice uniform error = 2/17

Uniform ORIGINAL epsilon over three := div(1, 3)

Uniform ORIGINAL epsilon over three = 1/3

Uniform ORIGINAL three term triangle upper bound := sum(2/17, 1/3)

Uniform ORIGINAL three term triangle upper bound = 23/51

source roles = [14, 4, 1/17, 1]

derived = [ASSUME f_N is L-Lipschitz on the WHOLE domain, ASSUME EVERY x:abs(f_N(x)-f(x))<=error_bound, 1/12]

source roles = [14, 4, 1/17, 1, 1/12]

derived = [abs(f(x)-f(y))<=abs(f(x)-f_N(x))+abs(f_N(x)-f_N(y))+abs(f_N(y)-f(y)), abs(x-y)<1/12 => bound<23/51, 是]

逐步反向核對

• S009 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。
• S008 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。
• S007 · premise_and_result_audit — 以精確有理數重算 Uniform ORIGINAL three term triangle upper bound，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 Uniform ORIGINAL epsilon over three，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 Uniform ORIGINAL twice uniform error，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 Uniform ORIGINAL epsilon over threeL，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 Uniform ORIGINAL three times Lipschitz constant，不以小數近似替代。
• S002 · premise_and_result_audit — 以精確有理數重算 Uniform ORIGINAL three times uniform error，不以小數近似替代。
• S001 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。

結論

(14, 4, 1/17, 1, 1/12, 23/51, 是, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)

依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。

本題覆核與來源綁定

覆核日期：2026-10-26

原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。