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Question 26321: A Discontinuous Sign Jump and Failure of Uniqueness Have Different Boundaries

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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← Question derivation index · Original question batch

R0051-26321 · q-20261015-advance-20261116-r02-intermediate-value-theorem-0096

Original question and givens

Use boundary mode=nonunique and positive exact scale=21. Give (mode, function_description, a, b, target, inclusive_bracket, multiplicity_statement, uniqueness_claim).

  • G001 — Use boundary mode=nonunique and positive exact scale=21. Give (mode, function_description, a, b, target, inclusive_bracket, multiplicity_statement, uniqueness_claim).

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.

Step-by-step derivation

S001 · implies

Before

source roles = [ivt-discontinuity-boundary]

After

derived = [[mode, function_description, a, b, target, inclusive_bracket, multiplicity_statement, uniqueness_claim]]

Conditions: none

List the full source field order without JavaScript.Original modes select contract6/4/6/6 and IVT8/7/8 roles,without padding or answer-shape inference.

Step identifiers and dependencies

Rule: source_ordered_fields_without_JS · Uses: G001

S002 · calculation

Before

IVT nonunique ORIGINAL scale square := mul(21, 21)

After

IVT nonunique ORIGINAL scale square = 441

Conditions: none

Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · implies

Before

source roles = [nonunique, 21]

After

derived = [nonunique, f(x)=x^2, -21, 21, 441, Yes, two_roots_possible, No]

Conditions: none

The jump takes only±s and misses0 despite endpoint bracketing because continuity fails.The scaled identity on[0,1] reaches at most s,not2s.For x² on[-s,s],target s² has two endpoint roots,so existence is not uniqueness.False roles follow the mode;do not pad a continuity field.

Step identifiers and dependencies

Rule: IVT_original_jump_nonbracket_nonunique_distinct_roles · Uses: G001, S002

Reverse checks

  • S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S002 · premise_and_result_audit — Recompute IVT nonunique ORIGINAL scale square 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.

Result

(nonunique, f(x)=x^2, -21, 21, 441, Yes, two_roots_possible, No)

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: 4e82a57e60ebc57f1e6e05314e7198a0ab3c9fa48ea7f3a2ed36303e34119443
  • candidate record SHA-256: 88eb7fa1f2a474e24899c1cfaa2616a8b69c13da22dab98f0871d17fe7902cea
  • semantic review receipt SHA-256: 90df1e889242e9afdf01e068788ee8500ef366ffcfc0496e30a6800de32631c3
  • question-first witness SHA-256: ae1e849608590ca856c9f3b6911ba58e3bf53c608eb7369606192a770f48bd5c
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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