# Question 22246: Continuity and Differentiability Apply on Different Interval Parts

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0043-22246 · q-20261014-advance-20261110-r02-mean-value-theorem-0021



## Original question and givens

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=1, B=10, C=401, a=-22, b=23 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (a, b, ordered_endpoints, closed_interval_continuity, open_interval_differentiability, certificate).


- G001 — Use the whole real polynomial f(t)=A*t^2+B*t+C with A=1, B=10, C=401, a=-22, b=23 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (a, b, ordered_endpoints, closed_interval_continuity, open_interval_differentiability, certificate).





## 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.


- [mko-mean-value-theorem](https://ocme.evemisslab.com/en/mko/mko-mean-value-theorem/)





## Step-by-step derivation

S001 · impliesBefore

source operation=mvt-hypothesis-audit;original snapshot and mathematics retained

After

explicit ordered source output fields=[a, b, ordered_endpoints, closed_interval_continuity, open_interval_differentiability, certificate]

Conditions: none

Retain the historical interface marker and list ordered source fields without JavaScript.MVT fields come from the original mode,without padding or answer-shape inference.Tags and fixed No markers retain their claim scope.

Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001





S002 · impliesBefore

ORIGINAL WHOLE f(t)=(1)t^2+(10)t+(401);a=-22<b=23;all endpoint[Ax,Axx,Bx,Axx+Bx,Axx+Bx+C] rows=[[-22, 484, -220, 264, 665], [23, 529, 230, 759, 1160]]

After

English source ALL endpoint intermediate SAFE guards passed;polynomial continuous on CLOSED[a,b] and differentiable on OPEN(a,b),not inferred from finite samples

Conditions: none

Finite sums/products preserve continuity and differentiability,with derivative2At+B.These are properties of the whole formula,not an inference from endpoint samples.SAFE guards qualify source numeric projection,not the mathematical domain of polynomials.

Step identifiers and dependenciesRule: mvt_whole_polynomial_hypotheses_and_SAFE_source_guard · Uses: G001, S001









## Reverse checks


- S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.





## Result

(-22, 23, Yes, Yes, Yes, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)

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.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 5a535e8d5a5cbb8f7a3ca044eb0925a28aee86824850d64e7fa1781d698e54fb
- candidate record SHA-256: 57514bf88547007ba9482b6950b4135adb9bb2353253bebc24a4324e44f318cd
- semantic review receipt SHA-256: 49cf6ec60211c92d1c0c302663af4d44479635dc8181fb61a20c3374288e21c5
- question-first witness SHA-256: 10fd25b4643e5ff12753daceaa61f9f2499bf381f65ef96e9d21185c7c0d0c5e
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0021.json)