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R0005-03365 · q-20261007-advance-20261008-r03-continuity-0040
The function is defined on all R. The left branch at x<2 is (3)x+(-5); the right branch at x>2 is (-3)x+(9). Its explicit center value is f(2)=1. Is it continuous at 2?
G001 — The function is defined on all R. The left branch at x<2 is (3)x+(-5); the right branch at x>2 is (-3)x+(9). Its explicit center value is f(2)=1. Is it continuous at 2?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=2
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
left branch slope times center := mul(3, 2)
After
left branch slope times center = 6
Conditions: none
Substitute the center a into this affine formula's product term.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
left branch center value := sum(6, -5)
After
left branch center value = 1
Conditions: none
Then add this branch's intercept,not the other branch's constant.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
right branch slope times center := mul(-3, 2)
After
right branch slope times center = -6
Conditions: none
Substitute the center a into this affine formula's product term.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
right branch center value := sum(-6, 9)
After
right branch center value = 3
Conditions: none
Then add this branch's intercept,not the other branch's constant.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
left L=1;right R=3;given f(a)=1
After
these are one-sided limits,not automatic point substitutions into f(a)
Conditions: none
For an affine branch the difference from its center limit L is m(x−a).For any ε>0 choose δ=ε/(1+|m|) to obtain the one-sided limit.The actual center value remains the separately specified v.
Rule: affine_one_sided_limits · Uses: G001, S005
Before
left limit minus point value := sub(1, 1)
After
left limit minus point value = 0
Conditions: none
Compute the left limit minus the actual point value.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
right limit minus point value := sub(3, 1)
After
right limit minus point value = 2
Conditions: none
Compare the right limit with the same actual point value.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
L−f(a)=0;R−f(a)=2
After
continuous at a=No
Conditions: none
In this real model with affine one-sided limits,continuity holds exactly when both limits and the point value agree.Equal left and right limits alone may still leave an incorrect center value.
Rule: two_sided_limit_equals_point · Uses: G001, S008
S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S008 · premise_and_result_audit — Recompute right limit minus point value with exact rational arithmetic, not rounded decimals.S007 · premise_and_result_audit — Recompute left limit minus point value with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S005 · premise_and_result_audit — Recompute right branch center value with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Recompute right branch slope times center with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute left branch center value with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute left branch slope times center 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.No
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.
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