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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'?2012-04-22
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    As in $f$ is a regulated function, i.e. it can always be approximated, as close as you like, by a step function. http://en.wikipedia.org/wiki/Regulated_function2012-04-22
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    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...2012-04-22

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