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

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

Statement refuted

Absolute continuity is preserved under arbitrary composition.

Facts & Assumptions

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

Counterexample

technique · direct

1.1givenalgebra

The derivative of $g$ is bounded on $(0, 1]$ and $g^{'} (0) = 0$ , so If $f$ is continuous on an interval $I$ and $∣ f^{'} ∣ \leq M$ at every interior point, then $∣ f (x) - f (y) ∣ \leq M ∣ x - y ∣$ for all $x, y \in I$ , so $f$ is Lipschitz with constant $M$ and uniformly continuous on $I$ and $C^{1}$ implies Lipschitz, Lipschitz implies absolutely continuous, and absolutely continuous implies continuous and bounded variation show that $g$ is absolutely continuous. Since $h^{'} (y) = 1 / (2 \sqrt{y})$ is integrable and $h (y) - h (0) = \int_{0}^{y} h^{'} (t) d t$ , Fundamental theorem of calculus for absolutely continuous functions shows that $h$ is absolutely continuous.

2.1step 1.1algebra

For $x > 0$ , $(h \circ g) (x) = ∣ x \sin (1 / x) ∣$ . On the alternating half-waves accumulating at $0$ , its total variation has a positive contribution comparable to $1 / n$ , and the harmonic series diverges. Thus $h \circ g$ is not of bounded variation.

3.1step 2.1∎

Every absolutely continuous function has bounded variation by $C^{1}$ implies Lipschitz, Lipschitz implies absolutely continuous, and absolutely continuous implies continuous and bounded variation, so $h \circ g$ is not absolutely continuous. Hence the universal composition assertion is false.

Depends on

Used by

A classical composition of absolutely continuous functions is not absolutely continuous Counterexample

Dependency tree · two levels

43 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)