Given a function $f$ that is differentiable at a point $c$, if we define (using the Riemann integral)
$$F(x) = \int_a^x f(t)dt.$$
I want to determine If $f$ is differentiable at $c$ then $F'$ is continuous at $c$.
This is apparently false, since I cannot guarantee that $F'$ exists in a neighborhood around $c$. However, I can't think of any counterexample to argue my idea, any suggestions?
Thanks