I'm trying to show that for an absolutely continuous function $f : \mathbb{R} \rightarrow \mathbb{R}$ with $f' \in L^{p}$, $1 < p < \infty$ it holds that \begin{align*} lim_{h \rightarrow 0} h^{1-p}(f(x+h)-f(x))^{p} = 0 \end{align*} for any real number $x$. This is trivial if $f'(x)$ exists, but I'm having trouble with the general case.
Using Hölder's inequality and the fact that $f' \in L^{p}$ I've been able to deduce that \begin{align*} \left\vert f(x+h)-f(x) \right\vert \leq \int_{x}^{x+h} \left\vert f'(t) \right\vert dt \leq h^{1/q} \cdot \left\Vert f' \right\Vert_{p} < \infty \end{align*} where $q$ is the Hölder conjugate of $p$. This then implies that \begin{align*} h^{1-p}\left\vert f(x+h)-f(x)) \right\vert^{p} \leq \left\Vert f' \right\Vert_{p}^{p} < \infty \end{align*} for all $h > 0$ and $x \in \mathbb{R}$, but beyond this I'm stuck. The bound above at least implies the existence of a convergent subsequence (if we replace $h$ by $1/n$ and $h \rightarrow 0$ with $n \rightarrow \infty$) but I'm not sure if this helps. I also had an idea that f' might be bounded a.e. on $[x,x+h]$, but I don't know how to prove it, or if it is even true. I would appreciate some hints to point me in the right direction.