Multiple of Absolutely Continuous Function is Absolutely Continuous

Theorem
Let $k$ be a real number.

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

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

Then $k f$ is absolutely continuous.

Proof
Note that if $k = 0$, then $k f$ is constant.

Hence, by Constant Real Function is Absolutely Continuous:


 * $k f$ is absolutely continuous if $k = 0$.

Take now $k \ne 0$.

Let $\epsilon$ be a positive real number.

Since $f$ is absolutely continuous, there exists real $\delta > 0$ such that for every set of disjoint closed real intervals $\closedint {a_1} {b_1}, \dotsc, \closedint {a_n} {b_n} \subseteq I$ with:


 * $\ds \sum_{i \mathop = 1}^n \paren {b_i - a_i} < \delta$

we have:


 * $\ds \sum_{i \mathop = 1}^n \size {\map f {b_i} - \map f {a_i} } < \frac \epsilon {\size k}$

Then:

whenever:


 * $\ds \sum_{i \mathop = 1}^n \paren {b_i - a_i} < \delta$

Since $\epsilon$ was arbitrary:


 * $k f$ is absolutely continuous if $k \ne 0$.

Therefore:


 * $k f$ is absolutely continuous for all $k \in \R$.