$\mathrm{lim}_{h\rightarrow 0} \int_0^h |f(x+h)-f(x)| dx=0$ almost everywhere

Question

I have the following exercises: given $f\in L^1(\mathbb{R})$, for almost all $x\in\mathbb{R}$ we have $\mathrm{lim}_{h\rightarrow 0} \int_0^h |f(x+t)-f(x)| dt=0$.

My idea was to show first that this holds for an indicator function, then generalizing approximating an integrable function with a sequence of simple functions, but it does not work. Any ideas on how to prove it?