# Absolutely Continuous Real Function is Continuous

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## Theorem

Let $I \subseteq \R$ be a real interval.

Let $f: I \to \R$ be an absolutely continuous real function.

Then $f$ is continuous.

## Proof

From Absolutely Continuous Real Function is Uniformly Continuous:

- $f$ is uniformly continuous.

Therefore, from Uniformly Continuous Real Function is Continuous:

- $f$ is continuous.

$\blacksquare$