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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False: every derivative is Riemann integrable

Statement

False claim: Every derivative on a compact interval is Riemann integrable.

Facts & Assumptions

Given: The false universal claim.

[L1]

$F (x) = x^{2} \sin (1 / x^{2})$ has an unbounded derivative whose Henstock–Kurzweil integral is $\sin 1$ ( $F (x) = x^{2} \sin (1 / x^{2})$ has an unbounded derivative whose Henstock–Kurzweil integral is $\sin 1$ ).

[L2]

Every derivative is Henstock–Kurzweil integrable (Every derivative is Henstock–Kurzweil integrable and satisfies Newton–Leibniz).

[L3]

The Darboux definition of Riemann integrability begins with a bounded function on $[a, b]$ (The lower and upper Darboux integrals of a bounded $f$ on $[a, b]$ as $\sup_{P} L (f, P)$ and $\inf_{P} U (f, P)$ , Darboux integrability as their equality, and the notation $\int_{a}^{b} f$ ).

Refutation

technique · contradiction

1.1

Suppose the claim were true; [L1] supplies a derivative unbounded near zero, while [L3] requires boundedness for Riemann integrability, a contradiction.

assume-contraL1L3

2.1

The correct conclusion for the same derivative is [L2]: it is HK integrable and [L1] evaluates its integral by endpoint difference.

L1L2discharge-contradiction∎

Remarks

The bounded Volterra derivative gives a stronger, distinct failure of Riemann integrability (Volterra's function is differentiable everywhere with bounded derivative, but its derivative is not Riemann integrable); it is not used in this refutation.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

25 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

Alessandro Fonda, The Kurzweil-Henstock Integral for Undergraduates, Ch. 1 (standard reference, not scraped)
Andrew Bruckner, Judith Bruckner and Brian Thomson, Real Analysis, Section 1.21 (standard reference, not scraped)