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Suppose we have a continuous function f:R→R. Suppose also that for a certain point c, lim(x→c)f′(x) exists. Must f′(c) exist as well, and be equal to this limit?

This isn't quite the same as asking if derivatives are always continuous. The well-known function f(x)=x^2*sin(1/x) is continuous and differentiable everywhere, but its derivative has no limit at x=0. I'm wondering if the derivative of a continuous function can have a discontinuity where its limit does exist.

I know this is a duplicate.. but the original question has a wrong answer. It assumed that f is differentiable (except at x=a) at an open interval which containing a .

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  • $\begingroup$ If $\lim_{x\to c}f'(x)$ exists, then by definition we must have that $f$ is differentiable on a deleted neighbourhood of $c$. Otherwise the expression doesn't even make sense and the limit isn't defined. $\endgroup$ – Jason Feb 18 '15 at 10:51
  • $\begingroup$ @Jason so you are saying that the assumptions are necessary? $\endgroup$ – Akkf Feb 18 '15 at 10:53

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