Linear Combination of Absolutely Continuous Function is Absolutely Continuous

Theorem

Let $\alpha, \beta$ be real numbers.

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

Let $f, g : I \to \R$ be absolutely continuous real functions.

Then $\alpha f + \beta g$ is absolutely continuous.

Proof

From Multiple of Absolutely Continuous Function is Absolutely Continuous, we have:

$\alpha f$ and $\beta g$ are absolutely continuous.

We therefore have, by Sum of Absolutely Continuous Functions is Absolutely Continuous:

$\alpha f + \beta g$ is absolutely continuous.

$\blacksquare$