False statementConstruction: Literature-sourcedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Absolute continuity is preserved under composition

Statement

If $F$ and $G$ are absolutely continuous, then $F \circ G$ is absolutely continuous.

Facts & Assumptions

Given: On $[0, 1]$ , let $g (0) = 0$ and $g (x) = x^{2} \sin^{2} (1 / x)$ for $x > 0$ , and let $h (y) = \sqrt{y}$ .

Refutation

technique · direct

1.1givenalgebra

The derivative of $g$ is bounded on $(0, 1]$ and $g^{'} (0) = 0$ , so $g$ is Lipschitz and AC. Also $h^{'} (y) = 1 / (2 \sqrt{y}) \in L^{1} [0, 1]$ , so $h$ is AC.

2.1step 1.1algebra

For $x > 0$ , $(h \circ g) (x) = ∣ x \sin (1 / x) ∣$ . Its successive alternating half-waves contribute a quantity comparable to $1 / n$ to total variation, so the variation diverges.

3.1step 2.1∎

Every AC function on a compact interval has bounded variation by $C^{1}$ implies Lipschitz, Lipschitz implies absolutely continuous, and absolutely continuous implies continuous and bounded variation. Hence $h \circ g$ is not AC, and the statement is false.

Depends on

$C^1$ implies Lipschitz, Lipschitz implies absolutely continuous, and absolutely continuous implies continuous and bounded variation

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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

Christopher Heil, Absolute Continuity and the Banach--Zaretsky Theorem, §3.2 (standard reference, not scraped)