TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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Every function of bounded variation is differentiable almost everywhere

Statement

Assume the Axiom of Countable Choice.

Every real-valued function of bounded variation on a compact interval is differentiable almost everywhere.

Facts & Assumptions

Given: Countable choice and a bounded-variation function $f : [a, b] \to R$ .

[A1]

The symbols are those of the statement.

Proof

technique · direct

1.1

By Jordan decomposition for functions of bounded variation, write $f = G - H$ with $G$ and $H$ increasing. The monotone differentiability theorem A monotone function is differentiable almost everywhere by the rising-sun route shows that both $G^{'}$ and $H^{'}$ exist almost everywhere.

given

2.1

On the common full-measure set where both derivatives exist, $f^{'} = G^{'} - H^{'}$ exists as well. Therefore $f$ is differentiable almost everywhere.

step 1.1

3.1

Steps 1.1 and 2.1 prove the theorem.

step 1.1step 2.1∎

Depends on

Used by

A BV function can fail continuity at one point and still be differentiable almost everywhere Example

Dependency tree · two levels

20 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

Richard F. Bass, Real Analysis for Graduate Students, Theorem 14.8 (standard reference, not scraped)