My problem is:
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a measurable function (w.r.t. the Lebesgue measure) that is in $L^2$. Show that the function
$F(x)=\int_0^x f(t)\,dt$
satisfies $|F(x)-F(y)| \leq C|x-y|^{\frac{1}{2}}$
I don't really know where to start. I feel like it's kind of because $f(t)$ isn't allowed to grow more quickly than $(x-t)^{-\frac{1}{2}}$ near $x$, but I don'tknow how to make anything out of this.