TITLE: Find derivative of $\lfloor{x}\rfloor$ in distribution
QUESTION [5 upvotes]: Find derivative in distribution of $f(x)=\lfloor{x}\rfloor=E(x)$
$$E(x)≤x≤E(x+1)$$

Answer is :

$$\lfloor{x}\rfloor '=\displaystyle\sum_{k=-\infty}^{\infty}\delta_{k}$$
I don't have any idea about how to.
Can you assist?
I'm too thankful

REPLY [5 votes]: Let $f(x)=\lfloor x\rfloor$. For any test function $\phi(x)$, we have
$$\begin{align}
\langle f',\phi\rangle&=-\langle f,\phi'\rangle\\\\
&=-\int_{-\infty}^\infty \lfloor x\rfloor \phi'(x)\,dx\\\\
&=-\sum_{n=-\infty}^\infty \int_{n}^{n+1} n\phi'(x)\,dx\\\\
&=-\sum_{n=-\infty}^\infty n (\phi(n+1)-\phi(n))\\\\
&=-\sum_{n=-\infty}^\infty n \phi(n+1)+\sum_{n=-\infty}^\infty n\phi(n)\\\\
&-\sum_{n=-\infty}^\infty (n-1) \phi(n)+\sum_{n=-\infty}^\infty n\phi(n)\\\\
&=\sum_{n=-\infty}^\infty \phi(n)\\\\
&=\sum_{n=-\infty}^\infty \langle \delta_n ,\phi\rangle\\\\
&= \langle \sum_{n=-\infty}^\infty\delta_n ,\phi\rangle
\end{align}$$
Hence, in distribution $$\lfloor x\rfloor' = \sum_{n=-\infty}^\infty \delta(x-n)$$as was to be shown!