I know that the following holds for Step function, but not sure if it holds for the distribution too.
Does the following hold for the Dirac delta distribution too?
$\delta$ is a linear functional from a space of test functions. The space is here taken from Schwartz space $S$ or the space of all smooth functions of compact support $D$. The Dirac distribution is differentiable everywhere $(-\infty, \infty)$ except at the point $x = 0$ where the function has a nontrivial jump discontinuity. This can be solved by removing the discontinuity of $\delta'$ by setting \begin{equation} \delta' = 0 \end{equation} which is continuous now on the entire line even though $\delta$ is not differentiable on the real line.