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if a function is differentiable for all real values of x, can i assume the derivative defined on all real values of x? I think the answer is yes but i couldn't think of based explanation. if it is so, why? thank you for your help.

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    $\begingroup$ What does differentiable mean anyway? $\endgroup$ – samjoe Jan 3 '18 at 14:58
  • $\begingroup$ Yes, as long as by "differentiable" you mean "has a finite derivative". If you're allowing for the existence of infinite (signed) derivatives, then of course this won't be the case, unless you're considering functions that take values to the extended real number line. $\endgroup$ – Dave L. Renfro Jan 3 '18 at 19:32
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You are right. The definition of an everywhere differentiable function is a function which has derivative at all points.

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If $x_0$ is a point in the domain of a function f, then f is said to be differentiable at $x_0$ if the derivative f ′($x_0$) exists.

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