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R0006-03938 · q-20261007-advance-20261009-r03-function-limit-0013
On a punctured real domain, f(x)=L+(13)(x−A) for x≠A. Given ε=13/3>0 and failed δ=26/39>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+(13)(x−A) for x≠A. Given ε=13/3>0 and failed δ=26/39>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.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 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.
Rule: punctured_limit_scope · Uses: G001
Before
counterexample displacement := div(13/3, 13)
After
counterexample displacement = 1/3
Conditions: none
Choose h=ε/|m|>0;division requires the nonzero slope.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
punctured output error := mul(13, 1/3)
After
punctured output error = 13/3
Conditions: none
At x=A+h,the output error |f(x)−L|=|m|h equals ε exactly.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
h=1/3>0;δ=2/3;h<δ=Yes;error=13/3;ε=13/3
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.
Rule: strict_delta_counterexample · Uses: G001, S003
Before
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.
Rule: bad_delta_not_no_limit · 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 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.(1/3, 13/3)
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: 199dec0470d3a35a127f475895b7e5b660cb72d8a14ae343b543add2b71fd9b9candidate record SHA-256: 0774be0ff883edb4f4143872daca7ecdef96194e43031afd18a583bec84c1b2esemantic review receipt SHA-256: 568eefd71709ac3dc16a0d5b3b4ad3e83376a8ef300c9e5493eee0721500b29dquestion-first witness SHA-256: d964b91bedb9d4918bbc055f567762a09f5a58af6c876268e719d5100332d855Lean: NOT_RUN · human review: NOT_PERFORMED