I need help proving the following extension: Suppose $f$ is integrable on $[a,b]$ and let $F(x) := \int_a^x f(x) dx$. If f is continuous at $x_0$, then $F$ is differentiable at $x_0$ and $F'(x_0) = f(x_0)$.
This is an extension of the theorem that states: Let $f$ be a continuous function on $[a,b]$ and let $F(x) = \int_a^x f(x) dx$, for $x \in (a,b)$. Then $F$ is differentiable at $x$ and $F'(x) = f(x)$.
(This theorem assumes $f$ is continuous on $[a,b]$. the extension assumes $f$ is continuous at a point.)