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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 $c$?

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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. – 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.

You can refer to Undergraduate Analysis by Lang for the proof of the generalized version of this result. There is an integral called the regulated integral and it is nicely developed in the book.

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