Explore related knowledge and questions →
← Question derivation index · Original question batch
R0005-03372 · q-20261007-advance-20261008-r03-continuity-0047
The function is defined on all R. The left branch at x<9 is (2)x+(-10); the right branch at x>9 is (-1)x+(17). Its explicit center value is f(9)=8. Is it continuous at 9?
G001 — The function is defined on all R. The left branch at x<9 is (2)x+(-10); the right branch at x>9 is (-1)x+(17). Its explicit center value is f(9)=8. Is it continuous at 9?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
left branch slope times center := mul(2, 9)
After
left branch slope times center = 18
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(18, -10)
After
left branch center value = 8
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(-1, 9)
After
right branch slope times center = -9
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(-9, 17)
After
right branch center value = 8
Conditions: none
Then add this branch's intercept,not the other branch's constant.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
left L=8;right R=8;given f(a)=8
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(8, 8)
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(8, 8)
After
right limit minus point value = 0
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)=0
After
continuous at a=Yes
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.Yes
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: b8e462c0e11ff6b6f793b81cdf06f8410160c15ed362fd58bf027ddc195c5914candidate record SHA-256: d0a0248bbd112a5c752e5ad0c7b14f9e832f85ef83f1ef75116f6d7198be8368semantic review receipt SHA-256: 5d74de11610ebc6037a4fc9946d77154d4ed18389f8f0feeb823148b6320c4c2question-first witness SHA-256: 1fba4ab0f13c2cfcc4d9fdebe492c442d43dd45700394a443daa28f333355ed9Lean: NOT_RUN · human review: NOT_PERFORMED