Let $f:[a,b]\to \mathbb{R}$ be a function of bounded variation. A well known result, known as Jordan's decomposition, states that we can write $f=W_1-W_2$ where $W_1,W_2$ are increasing and minimal in a certain sense. Now, these $W_i$ share some of the properties of $f$: for example, if $f$ is absolutely continuous, so will be $W_1$,$W_2$ (this is easy to see, just notice that $W_1(x)=\int_a^x f'^+$, $W_2=W_1-f$). My question is:
Suppose $f$ is differentiable (if needed, suppose also that $f'\in L^1([a,b])$, which implies that $f$ is absolutely continuous) at every point of the domain and that it is of bounded variation. Is it true that $W_1,W_2$ are everywhere differentiable?
I am quite sure that the answer is, in general, no, but I can't seem to find a $f$ patological enough.