Differentiability of an integral function at breakpoints

Let $f:[0,3]\to\mathbb R$ be defined by

$$f(x)=\begin{cases}0, & 0\le x<1,\ e^{x^2}-e, & 1\le x<2,\ e^{x^2}+1, & 2\le x\le3.\end{cases}$$

Now, define $F:[0,3]\to\mathbb R$ by $F(0)=0$ and $F(x)=\int_0^xf(t)\,dt$, for $0<x\le3$. Then