TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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Fundamental theorem of calculus for absolutely continuous functions

Assume the Axioms of Countable Choice and Dependent Choice. For $F : [a, b] \to R$ , the following are equivalent:

$F$ is absolutely continuous.
$F^{'}$ exists almost everywhere, $F^{'} \in L^{1} [a, b]$ , and $F (x) - F (a) = \int_{a}^{x} F^{'} (t) d t for every x \in [a, b] .$

Given: Countable choice, dependent choice, and a real function $F$ on the compact interval $[a, b]$ .

technique · direct

1.1given

2.1step 1.1algebra

The corollary The indefinite integral of an $L^{1}$ function is absolutely continuous makes $I_{f}$ AC, so $H : = F - F (a) - I_{f}$ is AC and has derivative zero a.e. The zero-derivative theorem makes $H$ constant; $H (a) = 0$ , giving (2).

3.1step 2.1∎

Conversely, (2) says $F = F (a) + I_{F^{'}}$ , and the same corollary makes it AC. The singleton interval satisfies both clauses directly.

Depends on