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R0005-03404 · q-20261007-advance-20261008-r03-continuity-0054
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.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.
Before
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.
Rule: continuity_model_scope · Uses: G001
Before
(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.
Rule: punctured_factor_cancellation · Uses: G001, S001
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
|(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.
Rule: extension_limit_epsilon_witness · Uses: G001, S003
Before
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.
Rule: extension_value_not_original_quotient · Uses: G001, S004
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.-18
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 7a82fffc619dd186a047a04759e142c1ff263ca2de3252c51f5b5074507db668candidate record SHA-256: 1d05c20bc813a6ffc3a912192f8358e7afce3002e01bde5ddc3a91cae439bdbfsemantic review receipt SHA-256: 5d74de11610ebc6037a4fc9946d77154d4ed18389f8f0feeb823148b6320c4c2question-first witness SHA-256: 1fba4ab0f13c2cfcc4d9fdebe492c442d43dd45700394a443daa28f333355ed9Lean: NOT_RUN · human review: NOT_PERFORMED