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Let $f$ be continuous on $(a,b)$ and let $c\in(a,b)$. Suppose $f'(x)$ exists for all $x\in(a,b)\backslash\{c\}$ and $\lim_{x\rightarrow c}f'(x)$ exists. Prove that $f'(c)$ exists.

I don't know what to do on this other than follow definitions. So for all $x\in(a,b)\backslash\{c\}$, the limit $$\lim_{y\rightarrow x}\frac{f(y)-f(x)}{y-x}$$ exists, and moreover, $$\lim_{x\rightarrow c}\lim_{y\rightarrow x}\frac{f(y)-f(x)}{y-x}$$ exists. To prove is that $$\lim_{y\rightarrow c}\frac{f(y)-f(c)}{y-c}$$ exists.


marked as duplicate by Hans Lundmark, Jonas Dahlbæk, Trevor Gunn, dantopa, Yujie Zha Jun 29 '17 at 15:57

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    $\begingroup$ See the first answer to this post. $\endgroup$ – David Mitra Jun 21 '13 at 15:35
  • $\begingroup$ This post has good answers as well (easily generalized). $\endgroup$ – David Mitra Jun 21 '13 at 15:46
  • $\begingroup$ @DavidMitra Thank you, nice proof. $\endgroup$ – PJ Miller Jun 21 '13 at 17:02

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