# Are all distributions differentiable?

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.

if $$T$$ is a distribution, its derivative with respect to the variable $$x$$ is the distribution $$T'$$ such that for any test function $$\varphi$$, $$T'(\varphi) = -T(\varphi')$$.
• Thanks for catching that: I started writing this using $\langle$ and $\rangle$ and then switched notation... – Robert Israel Mar 9 '20 at 19:05