Let $I\subset\mathbb R$ be an interval, and let $f:I\to\mathbb C$ be Lipschitz. Then $f$ is absolutely continuous on $I$ in the following sense: for every $\varepsilon>0$ there exists $\delta>0$ such that, for every $m\in\mathbb N$ and every finite pairwise disjoint family of subintervals $[a_k,b_k]\subset I$, $1\le k\le m$, with $a_k\le b_k$ and

\begin{align*} \sum_{k=1}^m (b_k-a_k)<\delta, \end{align*}

one has

\begin{align*} \sum_{k=1}^m |f(b_k)-f(a_k)|<\varepsilon. \end{align*}