Let $f \in H^1((0,\infty))$ where $H^1$ is the Sobolev space of functions $f \in L^2$ such that $f' \in L^2$ as well.
How can I compute
$$\lim_{h \downarrow 0}\frac{1}{h} \int_{h}^{x+h} (f(t-h)-f(t))dt?$$
Intuitively the answer should be
$$-\int_0^x f'(t) dt$$
but I do not know how to show it rigorously.