Alphabeta Math
False statementConstruction: Literature-sourcedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06
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  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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A continuous function of bounded variation is absolutely continuous

Statement

Every continuous function of bounded variation on a compact interval is absolutely continuous.

Facts & Assumptions

Given: Countable Choice, the standard Cantor--Lebesgue function C:[0,1]→[0,1], and the ternary Cantor set K.

Refutation

technique · direct
1.1

The function C is continuous and nondecreasing, hence has bounded variation, and C(K)=[0,1].

given
2.1

It maps the null Cantor set onto [0,1], so it fails property (N).

step 1.1
3.1

Absolute continuity would imply (N) by Absolutely continuous functions have Luzin's property (N). Therefore C is a continuous BV function that is not AC, refuting the statement.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources