# Question 27025: A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer [Return to catalogue page 136](https://ocme.evemisslab.com/en/questions/page/136/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations/#q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0075) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0053-27025 · q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0075 ## Original question and givens Fix N=25, a positive Lipschitz bound L=5, a whole-domain uniform error bound E=1/28≥0 and epsilon=1>3E. Give (N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, certificate). - G001 — Fix N=25, a positive Lipschitz bound L=5, a whole-domain uniform error bound E=1/28≥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](https://ocme.evemisslab.com/en/mko/mko-uniform-limit-continuity-theorem/) ## Step-by-step derivation S001 · impliesBefore source roles = [uniform-continuity-transfer] After derived = [[N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, certificate]] Conditions: none List the complete source field order and mode-specific roles without JavaScript or answer-shape inference;No,undefined and infinity retain their distinct meanings. Step identifiers and dependenciesRule: source_ordered_fields_without_JS · Uses: G001 S002 · calculationBefore Uniform ORIGINAL three times uniform error := mul(3, 1/28) After Uniform ORIGINAL three times uniform error = 3/28 Conditions: none Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore Uniform ORIGINAL three times Lipschitz constant := mul(3, 5) After Uniform ORIGINAL three times Lipschitz constant = 15 Conditions: none Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore Uniform ORIGINAL epsilon over threeL := div(1, 15) After Uniform ORIGINAL epsilon over threeL = 1/15 Conditions: none Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore Uniform ORIGINAL twice uniform error := mul(2, 1/28) After Uniform ORIGINAL twice uniform error = 1/14 Conditions: none Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore Uniform ORIGINAL epsilon over three := div(1, 3) After Uniform ORIGINAL epsilon over three = 1/3 Conditions: none Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore Uniform ORIGINAL three term triangle upper bound := sum(1/14, 1/3) After Uniform ORIGINAL three term triangle upper bound = 17/42 Conditions: none Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · impliesBefore source roles = [25, 5, 1/28, 1] After derived = [ASSUME f_N is L-Lipschitz on the WHOLE domain, ASSUME EVERY x:abs(f_N(x)-f(x))<=error_bound, 1/15] Conditions: none Fix the same N.The WHOLE uniform error and L-Lipschitz property of f_N are GIVEN premises,not inferred from a scalar N,samples or grids.L>0 and epsilon>3error give delta>0. Step identifiers and dependenciesRule: Uniform_GIVEN_same_fN_WHOLE_error_and_Lipschitz_premises · Uses: G001, S007 S009 · impliesBefore source roles = [25, 5, 1/28, 1, 1/15] After 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)<1/15 => bound<17/42, Yes] Conditions: none For arbitrary x,y with |x-y|