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I am learning about differentiability of functions and came to know that a function at sharp point is not differentiable.
For eg. $$f(x)=|x|$$ I could find out that $f(x)$ is not differentiable at $x=0$ because
$$\lim_{x\to 0^-}f'(x) \ne \lim_{x\to 0^+}f'(x) $$ This is all mathematical but I couldn't understand where the sharp point plays its role here ?
How sharp point makes these limits to evaluate different ?

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    **Very** informally, $f$ is differentiable at $a$ if a very very tiny bug sitting at $a$ can believe that the curve is flat at $a$, that its world is a straight line. The bug, sitting at $0$ on $y=|x|$, will not believe that the world is flat. Particularly if you turn the curve upside down, so that the "sharp" point is digging into its sitting end.2012-07-04
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    +1: You have a very imaginative vision of the life of bugs , @André!2012-07-04
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    A variant of this always gets a laugh. And maybe they actually will remember.2012-07-04

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