# Question 3927: Exact Punctured Counterexample to a Failed δ

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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.

R0006-03927 · q-20261007-advance-20261009-r03-function-limit-0002



## Original question and givens

On a punctured real domain, f(x)=L+(-2)(x−A) for x≠A. Given ε=2/2>0 and failed δ=6/4>0 with |m|δ>ε, give (h,error) for the specified positive witness h=ε/|m|. It must satisfy 0<h<δ but fail |f(A+h)−L|<ε. Use reduced positive-denominator fractions, retaining /1 for integers.


- G001 — On a punctured real domain, f(x)=L+(-2)(x−A) for x≠A. Given ε=2/2>0 and failed δ=6/4>0 with |m|δ>ε, give (h,error) for the specified positive witness h=ε/|m|. It must satisfy 0<h<δ but fail |f(A+h)−L|<ε. Use reduced positive-denominator fractions, retaining /1 for integers.





## 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](https://ocme.evemisslab.com/en/mko/mko-function-limit/)





## Step-by-step derivation

S001 · impliesBefore

declared real model with an excluded center for limit data

After

point value,legal off-center evaluation and limit are distinct

Conditions: none

A punctured limit does not read the center value,and evaluation at one legal point is not a limit.These declared affine or factorizable models do not represent arbitrary functions.

Step identifiers and dependenciesRule: punctured_limit_scope · Uses: G001





S002 · calculationBefore

counterexample displacement := div(1, 2)

After

counterexample displacement = 1/2

Conditions: none

Choose h=ε/|m|>0;division requires the nonzero slope.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001





S003 · calculationBefore

punctured output error := mul(2, 1/2)

After

punctured output error = 1

Conditions: none

At x=A+h,the output error |f(x)−L|=|m|h equals ε exactly.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · impliesBefore

h=1/2>0;δ=3/2;h<δ=Yes;error=1;ε=1

After

legal punctured input but strict output inequality fails

Conditions: none

The condition0<h<δ places the witness inside the punctured neighborhood;error equal to ε already violates strict error<ε,not merely error≤ε.

Step identifiers and dependenciesRule: strict_delta_counterexample · Uses: G001, S003





S005 · impliesBefore

this particular proposed δ is invalid

After

not a proof that the affine limit fails to exist

Conditions: none

Only this proposed δ fails;the affine model still allows the smaller δ=ε/(1+|m|).One bad choice does not disprove existence of the limit.

Step identifiers and dependenciesRule: bad_delta_not_no_limit · Uses: G001, S004









## Reverse checks


- S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S003 · premise_and_result_audit — Recompute punctured output error with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Recompute counterexample displacement with exact rational arithmetic, not rounded decimals.
- S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.





## Result

(1/2, 1/1)

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.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: cff439d3cadd3f74881319091062000266d3f8fa8bb43122dc4f98cb4e7693b4
- candidate record SHA-256: e1d8567e9f77a1692a92b7d8d2c5788299e8af1e9d4b39c7f9ef42d4c09df3a7
- semantic review receipt SHA-256: 568eefd71709ac3dc16a0d5b3b4ad3e83376a8ef300c9e5493eee0721500b29d
- question-first witness SHA-256: d964b91bedb9d4918bbc055f567762a09f5a58af6c876268e719d5100332d855
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261007-advance-20261009-r03-function-limit-0002.json)