Question 32297: Affine δ Witness with Strict Punctured Interior

Original question and givens

For arbitrary real A,L, define f(x)=L+(-21)(x−A) for x≠A on R∖{A}. With ε=1/2>0 and δ=2/42>0, does 0<|x−A|<δ guarantee |f(x)−L|<ε?

• G001 — For arbitrary real A,L, define f(x)=L+(-21)(x−A) for x≠A on R∖{A}. With ε=1/2>0 and δ=2/42>0, does 0<|x−A|<δ guarantee |f(x)−L|<ε?

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-function-limit

Step-by-step derivation

source roles = [qf-function-limit-affine-delta, affine_delta_validity, punctured_reals, []]

derived = [[nonstrict_budget_YesNo_STRING]]

Delta ORIGINAL absolute slope times delta numerator := mul(21, 2)

Delta ORIGINAL absolute slope times delta numerator = 42

Delta ORIGINAL left cross product times epsilon denominator := mul(42, 2)

Delta ORIGINAL left cross product times epsilon denominator = 84

Delta ORIGINAL epsilon numerator times delta denominator := mul(1, 42)

Delta ORIGINAL epsilon numerator times delta denominator = 42

source roles = [-21, 1, 2, 2, 42]

derived = [1/2, 1/21, 84, 42, No]

Delta counterexample ORIGINAL epsilon over nonzero absolute slope := div(1/2, 21)

Delta counterexample ORIGINAL epsilon over nonzero absolute slope = 1/42

Delta counterexample ORIGINAL delta plus threshold := sum(1/21, 1/42)

Delta counterexample ORIGINAL delta plus threshold = 1/14

Delta counterexample ORIGINAL midpoint distance := div(1/14, 2)

Delta counterexample ORIGINAL midpoint distance = 1/28

Delta counterexample ORIGINAL absolute slope times distance := mul(21, 1/28)

Delta counterexample ORIGINAL absolute slope times distance = 3/4

Delta counterexample ORIGINAL delta minus distance := sub(1/21, 1/28)

Delta counterexample ORIGINAL delta minus distance = 1/84

Delta counterexample ORIGINAL error minus epsilon := sub(3/4, 1/2)

Delta counterexample ORIGINAL error minus epsilon = 1/4

source roles = [-21, 1/2, 1/21]

derived = [1/42, 1/28, 3/4, 1/84, 1/4, choose x = A+r]

Reverse checks

• S012 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S011 · premise_and_result_audit — Recompute Delta counterexample ORIGINAL error minus epsilon with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute Delta counterexample ORIGINAL delta minus distance with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute Delta counterexample ORIGINAL absolute slope times distance with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Delta counterexample ORIGINAL midpoint distance with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Delta counterexample ORIGINAL delta plus threshold with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute Delta counterexample ORIGINAL epsilon over nonzero absolute slope with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S004 · premise_and_result_audit — Recompute Delta ORIGINAL epsilon numerator times delta denominator with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute Delta ORIGINAL left cross product times epsilon denominator with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute Delta ORIGINAL absolute slope times delta numerator with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.

Result

No

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.