Let $G(x)=\int_a^xg(t)dt$ for some $g(t):[a,b]\rightarrow \mathbb{R}$ ($g(t)$ is Riemann integrable on $[a,b]$) , and suppose we know that $G(x)$ is differentiable on $[a,b]$. Does it follow that $G'(x)$ is Riemann integrable on $[a,b]$?
I believe this is true, but I'm completely at a loss on how to prove it, and would really appreciate any hints.
Edit: I would like to know if this is true even if $g$ is discontinuous (but still integrable). The continuous case is clear from the fundamental theorem of calculs