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

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Lipschitz characterisation within absolutely continuous functions

Statement

Assume the Axioms of Countable Choice and Dependent Choice. For $F \in A C [a, b]$ and $L \geq 0$ , $F$ is $L$ -Lipschitz if and only if $∣ F^{'} ∣ \leq L$ almost everywhere.

Facts & Assumptions

Given: Countable choice, dependent choice, $F \in A C [a, b]$ , and $L \geq 0$ .

Proof

technique · direct

1.1given

If $F$ is $L$ -Lipschitz, every difference quotient is bounded by $L$ ; wherever the derivative exists, $∣ F^{'} ∣ \leq L$ . Absolute continuity itself follows from $C^{1}$ implies Lipschitz, Lipschitz implies absolutely continuous, and absolutely continuous implies continuous and bounded variation.

1.2givenalgebra

Conversely, the sharp FTC gives $F (y) - F (x) = \int_{x}^{y} F^{'}$ , whose absolute value is at most $L ∣ y - x ∣$ .

2.1step 1.1step 1.2∎

This proves both directions; it includes $L = 0$ and $a = b$ .

Depends on

Used by

One-sided semiconvex characteristics give W2 uniqueness and stability Lemma

Dependency tree · two levels

32 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, §3.5 (standard reference, not scraped)