Let $f$ be a function of bounded variation on $[a,b]$. I know that $$f = f_+ - f_-,$$ where $f_+$ and $f_-$ are both increasing, and that one such decomposition is obtained by taking $$f_{\pm}(x) = \frac{1}{2}(T_f(a,x) \pm f(x),$$ where $T_f(a,x)$ is the total variation of $f$ on $[a,x]$.
Since $f_{\pm}$ are increasing, $f_{\pm}'$ exist almost everywhere. I was told that $f_+'$ and $f_-'$ have disjoint supports, i.e. if $f_+'(x) > 0$, then $f_-'(x) = 0$. However, I'm having a hard time proving this. I tried splitting it up based on whether $f'(x)$ was positive, hoping to get some sign contradiction, but this didn't work.