Is it true that if $\frac{d}{dx}f(x)$ is continuous, then $f(x)$ is continuous too?
If not, can you give a counterexample?
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$\begingroup$ Have you tried relating the definition of the derivative to the definition of continuity? $\endgroup$– Mark BennetJun 8, 2013 at 14:14
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$\begingroup$ Do you mean the derivative in the sense of distributions? $\endgroup$– SiméonJun 8, 2013 at 14:19
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2 Answers
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Just the fact that your function $f(x)$ is differentiable is enough to prove that it is continuous. The derivative $\frac{d}{dx}f(x)$, need not even be continuous. Please have a look here http://www-math.mit.edu/~djk/18_01/chapter02/proof04.html
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To be differentiable at a point $a$, a function must also be continuous at that point $a$. In your question, this holds for all $a\in \mathbb{R}$.
