The second part of the fundamental theorem of calculus is stated in wipedia (http://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus#Second_part) as:
"Let $f$ and $F$ be real-valued functions defined on a closed interval $[a, b]$ such that the derivative of $F$ is $f$. That is, $f$ and $F$ are functions such that for all $x$ in $[a, b]$, $F'(x) = f(x).$ If f is Riemann integrable on $[a, b]$ then $\int_a^b f(x)\,dx = F(b) - F(a).$ The Second part is somewhat stronger than the Corollary because it does not assume that $f$ is continuous."
Can anyone give an example where the second part can be applied, but not the first? ($f$ not continuous). My intuition says that when $F$ is differentiable everywhere on $[a, b]$, its derivative is continuous so such an example cannot exist.