Alphabeta Math
False statementConstruction: Literature-sourcedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06
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Luzin's property (N) implies absolute continuity

Statement

A function with Luzin's property (N) is absolutely continuous.

Facts & Assumptions

Given: Define F(0)=0 and F(x)=xsin⁡(1/x) for 0<x≤1.

Refutation

technique · direct
1.1

The function F is C1 on every interval [1/(n+1),1/n], hence Lipschitz there. These intervals and the singleton {0} form a countable cover, so F maps every null set to a null set and has property (N) as in Luzin's property (N) on a compact interval.

givenalgebra
2.1

Since F′(x)=sin⁡(1/x)−cos⁡(1/x)/x for x>0, alternating subintervals give infinite variation. Thus F is not BV.

step 1.1algebra
3.1

Every absolutely continuous function on a compact interval is BV by C1 implies Lipschitz, Lipschitz implies absolutely continuous, and absolutely continuous implies continuous and bounded variation. Hence this property-(N) function is not AC.

step 2.1∎

Depends on

Used by

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Dependency tree · two levels

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Sources