Question 27017: A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer

Original question and givens

Fix N=17, a positive Lipschitz bound L=2, a whole-domain uniform error bound E=1/20≥0 and epsilon=1>3E. Give (N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, certificate).

• G001 — Fix N=17, a positive Lipschitz bound L=2, a whole-domain uniform error bound E=1/20≥0 and epsilon=1>3E. Give (N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, certificate).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-uniform-limit-continuity-theorem

Step-by-step derivation

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, 1/20)

Uniform ORIGINAL three times uniform error = 3/20

Uniform ORIGINAL three times Lipschitz constant := mul(3, 2)

Uniform ORIGINAL three times Lipschitz constant = 6

Uniform ORIGINAL epsilon over threeL := div(1, 6)

Uniform ORIGINAL epsilon over threeL = 1/6

Uniform ORIGINAL twice uniform error := mul(2, 1/20)

Uniform ORIGINAL twice uniform error = 1/10

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

Uniform ORIGINAL epsilon over three = 1/3

Uniform ORIGINAL three term triangle upper bound := sum(1/10, 1/3)

Uniform ORIGINAL three term triangle upper bound = 13/30

source roles = [17, 2, 1/20, 1]

derived = [ASSUME f_N is L-Lipschitz on the WHOLE domain, ASSUME EVERY x:abs(f_N(x)-f(x))<=error_bound, 1/6]

source roles = [17, 2, 1/20, 1, 1/6]

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)<1/6 => bound<13/30, Yes]

Reverse checks

• S009 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
• S008 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
• S007 · premise_and_result_audit — Recompute Uniform ORIGINAL three term triangle upper bound with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute Uniform ORIGINAL epsilon over three with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Uniform ORIGINAL twice uniform error with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Uniform ORIGINAL epsilon over threeL with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute Uniform ORIGINAL three times Lipschitz constant with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute Uniform ORIGINAL three times uniform error with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.

Result

(17, 2, 1/20, 1, 1/6, 13/30, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.