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My question is quite similar to the following question Are distributions all continuous? but Im still not very clear on the answer provided. I know that for a function to be differentiable, it must be continuous, and according to the given answer, NOT all distributions are continuous, but are they all differentiable?? I would really appreciate a simple explanation and example, thanks in advance.

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The source of your confusion is that two very different definitions of "differentiation" are being used here. Even when they can be considered as ordinary functions, distributions are in general not differentiable as functions. However, they are differentiable as distributions. The definition of the derivative of a distribution is:

if $T$ is a distribution, its derivative with respect to the variable $x$ is the distribution $T'$ such that for any test function $\varphi$, $ T'(\varphi) = -T(\varphi')$.

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  • $\begingroup$ Thanks for catching that: I started writing this using $\langle$ and $\rangle$ and then switched notation... $\endgroup$ – Robert Israel Mar 9 '20 at 19:05
  • $\begingroup$ There is an extra layer of confusion in the question beyond this, which is that the notion of "continuity" referred to in the linked question is entirely separate from the notion of continuity of functions (when you think of distributions as "generalized functions") and distributions are usually defined to always be continuous in that sense (contrary to the definition in the linked question). $\endgroup$ – Eric Wofsey Mar 9 '20 at 19:13
  • $\begingroup$ I see, so if I may, does that mean all distributions are differentiable as distributions ? thanks $\endgroup$ – curiousboi Mar 10 '20 at 4:58
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    $\begingroup$ Yes, that's right. $\endgroup$ – Robert Israel Mar 10 '20 at 11:44
  • $\begingroup$ Okay I see, so if I'm not mistaken the derivative of a distribution T that u have laid out in your answer is the definition of the weak derivative yes? If that is so, my question is why are all distributions differentiable, at least in the sense of the weak derivative? thank you @Robert Israel $\endgroup$ – curiousboi Mar 10 '20 at 14:35

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