Definition 1: A function $f:[a,b]\to\mathbb{R}$ is piecewise $C^1$, shortly $f\in KC^1([a,b])$, if exists a partition $a=t_0<t_1<\ldots<t_n=b$ s.t. $f|_{(t_{k-1},t_k)}\in C^1([t_{k-1},t_k])$, for $k=1,\ldots,n$.
Definition 2: A function $f:[a,b]\to\mathbb{R}$ is absolutely continuous, shortly $f\in AC([a,b])$, if for every $\epsilon>0$ exists $\delta$ s.t. if $\{[a_k,b_k]\}_{k=1}^n$, is a finite collection of disjoint intervals in $[a,b]$, it holds $$ \sum_{k=1}^n (b_k-a_k)<\delta\quad\Rightarrow\quad\sum_{k=1}^n |f(b_k)-f(a_k)|<\epsilon $$ Moreover, an important result states that $f\in AC([a,b])$ iff
- $f$ derivable in $[a,b]$ and
- $f(x)=f(a)+\int_a^xf'(t)dt\;$ for every $x\in[a,b]$
It is easy to prove that $KC^1[a,b]\subset AC[a,b]$ (i.e. every piecewise $C^1$ function is also absolutely continuous), but I can't find an exemple of a function $f\in AC([a,b])\setminus KC([a,b])$, so an absolutely continuous function which is not piecewise $C^1$.
Can anyone provide an example?