I'm working on a proof of the following theorem:
"Let $f\colon [a,b]\to\mathbb{R}$ be monotone increasing, $F\colon [a,b]\to\mathbb{R}$ such that $F(x):=\int_{[a,x]} f$ and $x_0 \in [a,b]$. Then $F$ is differentiable at $x_0$ if and only if $f$ is continuous at $x_0$."
Now, the leftward implication follows easily from the First Fundamental Theorem of Calculus, but I'm having difficulties proving the rightward one; I've tried proof by contradiction but without much success for now. So, I'd appreciate any hint about how to prove this remaining implication.
Best regards,
lorenzo.