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

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Every absolutely continuous function is Lipschitz

Every absolutely continuous function on a compact interval is Lipschitz.

Given: $F (x) = \sqrt{x}$ on $[0, 1]$ .

technique · direct

1.1

Since $F^{'} (x) = 1 / (2 \sqrt{x})$ belongs to $L^{1} [0, 1]$ , $F (x) = \int_{0}^{x} F^{'}$ is absolutely continuous.

given

2.1

If $F$ were $L$ -Lipschitz, then $\sqrt{x} \leq L x$ for $x > 0$ , hence $1 / \sqrt{x} \leq L$ , which fails as $x ↓ 0$ .

step 1.1algebra

3.1

Therefore $F$ is AC but not Lipschitz.

step 2.1∎

Used by

Nothing in the library uses this result yet.

Nothing. This result depends on no other item in the library.