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I want to know that if a differentiable function is differentiated, it is still differentiable.

I mean, let $f \ $ be a differentiable function, and $f' \ $ its derivative. Is $f' \ $ always also a differentiable function?

Blue
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Sigrid
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    Not necessarily. $f'$ does not even need to be continuous, let alone differentiability. – Koro Jun 28 '21 at 03:58
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    Is there some reason you think it would be? Your insight and investigation could be very valuable! – A rural reader Jun 28 '21 at 03:58
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    Take a function that isn't differentiable. Integrate it. You now have a function that is differentiable, with nondifferentiable derivative. Simples! – Gerry Myerson Jun 28 '21 at 04:14
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    See this great answer by Dave Renfro to the question How discontinuous can a derivative be? Spoiler alert: very discontinuous, for numerous definitions of "very". Btw, it's worth mentioning that for functions of a complex variable, differentiable (on, say, an open set) does imply that the derivative is differentiable, hence complex differentiable functions are infinitely differentiable, and even better, they are analytic (can be expressed at least locally as a power series). –  Jun 28 '21 at 05:11
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    Mathematicians care a good amount about how differentiable a function is, so much that there is a definition: a function $f$ is of class $C^r$ if it is $r$th order continuously differentiable; i.e., if its $r$th derivative exists and is continuous. We call $C^\infty$ functions smooth. – While I Am Jun 28 '21 at 12:34

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