Suppose you have an absolutely continuous function $f$, with derivative $f'\in L^p(\mathbb R)$ for some $p>1$. Then I would like to show that there exist constants $L$ and $\alpha$ such that
$|f(x)-f(y)| \leq L |x-y|^{\alpha}, \forall x,y.$
Since $f$ is absolutely continuous, we have that $f(x) - f(y) = \int_y^x f'(t) dt$. Then I should maybe use Hölder inequality, but I don't know how to apply it in this case. Any help would be appreciated!
Thanks!