For arbitrary real A,L, define f(x)=L+(8)(x−A) for x≠A on R∖{A}. With ε=3/1>0 and δ=3/16>0, does 0<|x−A|<δ guarantee |f(x)−L|<ε?
G001 — For arbitrary real A,L, define f(x)=L+(8)(x−A) for x≠A on R∖{A}. With ε=3/1>0 and δ=3/16>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.
BASE sources do not acquire a generic legacy operation/domain or unknown-metadata ban.Retain metadata WITHOUT using it as an answer oracle.PULSE/legacy enforce their exact source keys,operation/domain.Do not impose universal tuples,/1 or input bounds.
Delta ORIGINAL absolute slope times delta numerator := mul(8, 3)
After
Delta ORIGINAL absolute slope times delta numerator = 24
Conditions: none
Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation.
Delta ORIGINAL left cross product times epsilon denominator := mul(24, 1)
After
Delta ORIGINAL left cross product times epsilon denominator = 24
Conditions: none
Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation.
Delta ORIGINAL epsilon numerator times delta denominator := mul(3, 16)
After
Delta ORIGINAL epsilon numerator times delta denominator = 48
Conditions: none
Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation.
FIRST validate ALL FOUR positive raw fraction components,EVEN at m0.Because the input distance is STRICTLY below δ,the budget criterion is|m|δ≤ε,NOT<.Equality is legal;m0 gives error0.The target is L,not undefined f(A);A/L are arbitrary fixed reals,not invented parameters.
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
Yes
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.