# Question 3422: Center Value for a Continuous Extension

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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.

R0005-03422 · q-20261007-advance-20261008-r03-continuity-0072



## Original question and givens

The original quotient (x²−(9)²)/(x−(9)) is defined only away from 9. Choose the center value of a new continuous extension to all R.


- G001 — The original quotient (x²−(9)²)/(x−(9)) is defined only away from 9. Choose the center value of a new continuous extension to all R.





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





## Step-by-step derivation

S001 · impliesBefore

declared real model;center a=9

After

distinguish punctured limits,point values and a new extension

Conditions: none

Use the declared real formulas;point values,punctured limits and values of a newly defined extension are different data.

Step identifiers and dependenciesRule: continuity_model_scope · Uses: G001





S002 · impliesBefore

(x²−(9)²)/(x−(9));x≠9

After

(x−(9))(x+(9))/(x−(9))=x+(9)

Conditions: none

Factor first and cancel x−a only because x≠a;the original quotient remains undefined at a.

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





S003 · calculationBefore

extension center value := sum(9, 9)

After

extension center value = 18

Conditions: none

On the punctured domain the quotient equals x+a,whose limit as x approaches a is a+a.

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





S004 · impliesBefore

|(x+a)−2a|=|x−a|;a=9

After

δ=ε>0 proves the punctured limit 18

Conditions: none

For any ε>0 choose δ=ε;0<|x−a|<δ makes the difference from2a smaller than ε.This is not finite sampling.

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





S005 · impliesBefore

new extension g equals the original quotient off a

After

continuity at a requires and is achieved by g(a)=18

Conditions: none

Uniqueness of the limit forces the new value2a;with that value,g equals x+a on all reals.This does not alter the original quotient's undefined value at a.

Step identifiers and dependenciesRule: extension_value_not_original_quotient · 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 extension center value with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.





## Result

18

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: fb65c61649f2e30ef27b62e5e1ff885b977625a5ed8ce727adf1ebbf2e950896
- candidate record SHA-256: de4d44a331456ae948f29c02de0d5bf4c39cc3d7b089cfeb86882fb6b17db72c
- semantic review receipt SHA-256: 5d74de11610ebc6037a4fc9946d77154d4ed18389f8f0feeb823148b6320c4c2
- question-first witness SHA-256: 1fba4ab0f13c2cfcc4d9fdebe492c442d43dd45700394a443daa28f333355ed9
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261007-advance-20261008-r03-continuity-0072.json)