# Existence of Derivative for an Integral

Let $f$ be a Riemann integrable function defined on $[-2,2]$. Define a function $F \colon (-1,1) \to \mathbb{R}$ by $$F(h)=\int_0^1 h | f(x+h)-f(x)|\, dx.$$ Show that the derivative $F'(0)$ exists.

I started form $\lim_{h\to 0}$ $\frac{F(h)-F(0)}{h}$

By definition, $F(0)=0$ since it $F(0)$ becomes a definite integral of zero.