Proof of continuity of the integral of a regulated function.

If $f:[a,b] \to \mathbb{R}$ is regulated, $F(x):= \int_c^x f$ for fixed $c \in (a,b)$, $F$ is differentiable at $c$ and $F'(c) = f(c)$. How would you prove that $f$ is continuous at $c$?

-
what do you mean by $f$ is 'regulated'? –  Beni Bogosel Apr 22 '12 at 14:35
As in $f$ is a regulated function, i.e. it can always be approximated, as close as you like, by a step function. en.wikipedia.org/wiki/Regulated_function –  user26069 Apr 22 '12 at 15:58
You really should include that in your question, because it is not a very used notion. At least I didn't heard of it until now... –  Beni Bogosel Apr 22 '12 at 17:29

Use the fact that $f$ is a uniform limit of step functions.