Use boundary mode=cusp_at_zero, a=-9, b=10, scale=9, offset=8. degenerate_interval declares a≥b; jump_at_zero and cusp_at_zero declare a<0<b and nonzero scale, with the named interior defect. Give (mode, a, b, scale, offset, closed_interval_continuity, open_interval_differentiability, theorem_applicability, boundary).
G001 — Use boundary mode=cusp_at_zero, a=-9, b=10, scale=9, offset=8. degenerate_interval declares a≥b; jump_at_zero and cusp_at_zero declare a<0<b and nonzero scale, with the named interior defect. Give (mode, a, b, scale, offset, closed_interval_continuity, open_interval_differentiability, theorem_applicability, boundary).
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.
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.
Zero is interior.A nonzero jump has different one-sided limits and is neither continuous nor differentiable.A cusp has opposite nonzero one-sided slopes:continuous but not differentiable.Adding an offset removes neither defect.The No withholds MVT application under these failed hypotheses.
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
(cusp_at_zero, -9, 10, 9, 8, Yes, No, No, open_interval_differentiability_fails_at_zero)
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.