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R0005-03396 · q-20261007-advance-20261008-r03-continuity-0096
On all R, The left branch at x<8 is (1)x+(20); the right branch at x>8 is (-3)x+(53). The center value is 8. Give (left limit−center value, right limit−center value).
G001 — On all R, The left branch at x<8 is (1)x+(20); the right branch at x>8 is (-3)x+(53). The center value is 8. Give (left limit−center value, right limit−center value).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=8
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(1, 8)
After
left branch slope times center = 8
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(8, 20)
After
left branch center value = 28
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, 8)
After
right branch slope times center = -24
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(-24, 53)
After
right branch center value = 29
Conditions: none
Then add this branch's intercept,not the other branch's constant.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
left L=28;right R=29;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(28, 8)
After
left limit minus point value = 20
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(29, 8)
After
right limit minus point value = 21
Conditions: none
Compare the right limit with the same actual point value.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
L−f(a)=20;R−f(a)=21
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.(20, 21)
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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