When $F(t)=\int_0^tf(s)ds$ is differentiable everywhere?

Question

Let $f:\mathbb{R}\to \mathbb{R}$ be a function that is continuous almost everywhere.

1) Is the function $F(t)=\int_0^tf(s)ds$ differentiable everywhere ?

2) What is the "weakest" condition on $f$ (in general) that makes $F(t)=\int_0^tf(s)ds$ differentiable everywhere ? $f$ continuous ?