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 $c$?
Proof of continuity of the integral of a regulated function.
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real-analysis
analysis
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0You really should include that in your question, because it is not a very used notion. At least I didn't heard of it until no$w$... – 2012-04-22
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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.