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

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A monotone function is differentiable almost everywhere by the rising-sun route

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Let $F : [a, b] \to R$ be monotone. Then $F$ is differentiable at Lebesgue-almost every point of $(a, b)$ .

Facts & Assumptions

Given: Countable choice and a monotone function $F : [a, b] \to R$ .

[A1]

The symbols are those of the statement.

Proof

technique · direct

1.1given

Replacing $F$ by $- F$ if necessary, we may assume that $F$ is nondecreasing. By A nondecreasing function splits uniquely into a jump part and a continuous part, write $F = J_{F} + C_{F}$ with $C_{F}$ continuous and nondecreasing. The continuous theorem A continuous monotone function is differentiable almost everywhere by the rising-sun route gives differentiability of $C_{F}$ almost everywhere, while A jump function has derivative zero almost everywhere gives $J_{F}^{'} = 0$ almost everywhere.

2.1step 1.1

On the common full-measure set where both derivatives exist, $F^{'} = C_{F}^{'} + J_{F}^{'} = C_{F}^{'}$ . Therefore $F^{'}$ exists almost everywhere.

A monotone function is differentiable almost everywhere by the rising-sun route

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources