This post is discussing the connection between this equation
\begin{equation} \hat{p}(x) = \frac{1}{m} \sum_{i=1}^m \delta(x - x^{(i)}) \tag{3.28} \end{equation}
and this equation
$\widehat {F}_{n}(t)={\frac {{\mbox{number of elements in the sample}}\leq t}{n}}={\frac {1}{n}}\sum _{i=1}^{n}\mathbf {1} _{X_{i}\leq t}$
This post says
the derivative of a step function is the Dirac delta.
Consider this step function,
\begin{equation} \hspace{50pt} u(x) = \left\{ \begin{array}{l l} 1 & \quad x \geq 0 \\ 0 & \quad \text{otherwise} \end{array} \right. \hspace{50pt} \end{equation}
Per this post, the function above is not differentiable at $=0$.
Function is constant in (0,∞) and (−∞,0).The derivative is therefore 0.
Is it reasonable to claim that the derivative of a step function is the Dirac delta?