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

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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A Lipschitz function after an absolutely continuous function is absolutely continuous

Statement

If $F \in A C [a, b]$ and $h$ is Lipschitz on $F ([a, b])$ , then $h \circ F \in A C [a, b]$ .

Facts & Assumptions

Given: $F \in A C [a, b]$ and a Lipschitz constant $L$ for $h$ on its image.

Proof

technique · direct

1.1givenchoose

For $ε > 0$ , use the AC modulus of $F$ from Absolute continuity on a compact interval with tolerance $ε / L$ when $L > 0$ .

2.1step 1.1algebra

Sum $∣ h (F (v_{j})) - h (F (u_{j})) ∣ \leq L ∣ F (v_{j}) - F (u_{j}) ∣$ over a short disjoint family.

3.1step 2.1∎

If $L = 0$ the composite is constant; otherwise step 2.1 is the AC condition.

Depends on

Absolute continuity on a compact interval

Used by

Dependency tree · two levels

6 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)