Absolutely Continuous Real Function is Continuous

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.

$\blacksquare$