If $f$ is differentiable at $c$ then $F$ is differentiable at $c$

Question

I've be having trouble with this question and am looking for some advice.

Let $a,b \in \mathbb{R}, a < b$. Let $f$ be an integrable function on $[a, b]$, let $c \in(a, b)$, and let $$F(x) =\int_0^x \ f(t)\,dt.$$ Prove or disprove the following:

If $f$ is differentiable at $c$ then $F$ is differentiable at $c$.

Answer 1

By the first fundamental theorem of calculus, $F'(c)=f(c)$. So in fact we don't even need $f$ to be differentiable at $c$; it is enough for $f$ to be continuous at $c$.