Let $f:[0,1] \rightarrow \mathbb{R}$ be Riemann integrable, and for $0 \leqslant x \leqslant 1$ set $F(x)=\int_{0}^{x} f(t) d t$.

Assuming that $f$ is continuous, prove that for every $0<x<1$ the function $F$ is differentiable at $x$, with $F^{\prime}(x)=f(x)$.

If we do not assume that $f$ is continuous, must it still be true that $F$ is differentiable at every $0<x<1$ ? Justify your answer.