CounterexampleConstruction: 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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A classical composition of absolutely continuous functions is not absolutely continuous

Statement refuted

The composition of two absolutely continuous functions must be absolutely continuous.

Facts & Assumptions

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

Counterexample

technique · direct

1.1

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

givenalgebra

2.1

$(h \circ g) (x) = ∣ x \sin (1 / x) ∣$ for $x > 0$ . On alternating half-waves its total variation has a positive contribution comparable to $1 / n$ ; the harmonic sum diverges.

step 1.1algebra

3.1

Thus the composite is not BV and hence not AC, proving the general failure announced in The composition of two absolutely continuous functions need not be absolutely continuous.

step 2.1∎

Depends on

The composition of two absolutely continuous functions need not be absolutely continuous

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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)