Are all distributions differentiable?

Question

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.

Answer 1

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')$.