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Hi guys Im really having problems with this question:

Suppose $g$ is continuous in $(-1,1)$ and differentiable in $(-1,0)\cup(0,1)$. Prove that if $\lim\limits_{x\to C^+}g'(x)=\lim\limits_{x\to C^-}g'(x)$ with both limits finite, then $g$ is differentiable at $0$.

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The image that you supplied seems to have limits as $x\to C^+$ and $x\to C^-$, so that’s the way I edited it, but I’d bet that those should be limits as $x\to 0^+$ and $x\to 0^-$. –  Brian M. Scott Apr 8 '12 at 11:20
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Hi mugool. If this is homework, you should put the homework tag. –  yohBS Apr 8 '12 at 11:31
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Please tell us what have you tried. Also, consider Brian's remark there--Brian should be right or the problem does not make much sense. –  user21436 Apr 8 '12 at 17:59
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1 Answer

Go to the definition of the derivate when the increment converges to zero from right side and after from left side. Apply in each case the mean value theorem (Lagrange) in the close interval from the point (0 in your case) to the point plus a value of the increment samall enough such the conditions are Ok for the theorem Lagrange.

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C is zero if I understood your question ? –  alpha.Debi Apr 8 '12 at 12:22
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