State and prove the Fundamental Theorem of Calculus.

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

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be differentiable, and set $g(x)=f^{\prime}(x)$ for $0 \leqslant x \leqslant 1$. Must the Riemann integral of $g$ from 0 to 1 exist?