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A shortcut given in a book says that to check differentiability of a continuous function at $x=a$ we can differentiate $f(x)$ to get $f'(x)$ and if $\lim _{x\to a} f'(x)$ exists then function is differentiable at $x=a$.

Will above shortcut always work or will it fail in some cases?

Mathematics
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  • The shortcut will work only if the both the limits $\lim_{x\to a^{-}} f'(x), \lim_{x\to a^{+}} f'(x) $ exist. If one or both of these limits do not exist then you can't draw any conclusions regarding existence of $f'(a) $. The shortcut is typically used when $f$ is defined in piecewise manner. – Paramanand Singh Nov 12 '17 at 14:27
  • @ParamanandSingh Is there any specific reason as why we can't draw any conclusion if one of the limit does not exist. – Mathematics Nov 12 '17 at 14:57
  • Well Mean Value Theorem allows you to conclude about left derivative from $\lim_{x\to a^{-}} f'(x) $ and similar thing happens for right derivative. If these exist and are equal you say that the derivative exist. If these exist and are unequal then you say derivative does not exist. But if one of these limits does not exist then it is not possible to draw conclusions. Moreover there are explicit examples where the limits don't exist and yet the derivative exists. Check $f(x) =x^{2}\sin(1/x),f(0)=0$ where the derivative $f'(0)=0$ but left/right limit of $f'(x) $ does not exist as $x\to 0$. – Paramanand Singh Nov 12 '17 at 15:05

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