Alphabeta Math
False statementConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-21
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  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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False: every Henstock–Kurzweil integrable function is bounded

Statement

False claim: Every Henstock–Kurzweil integrable function on a compact interval is bounded.

Facts & Assumptions

Given: The false universal claim.

[L1]

F(x)=x2sin⁡(1/x2) has an unbounded derivative whose Henstock–Kurzweil integral is sin⁡1 (F(x)=x2sin⁡(1/x2) has an unbounded derivative whose Henstock–Kurzweil integral is sin⁡1).

Refutation

technique · contradiction
1.1

Suppose the claim were true; the derivative in [L1] is HK integrable and unbounded on the same compact interval.

assume-contraL1
2.1

This contradicts the claimed boundedness, so the claim is false.

step 1.1discharge-contradiction∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources