How to "invert" the argument of the Heaviside function?

Question

How can I go from $\theta (x-x_0)$ to $\theta(x_0 - x)$ analytically?. In the first, the Heaviside function gives 1 when $x > x_0$. I need it to give $1$ when $x < x_0$. Do I explain myself? (1st question in this forum).

Answer 1

If your definition is $$ \delta(x)=\begin{cases}1 & \text{if } x>0,\\ 0 & \text{if } x\le 0,\end{cases} $$ then $$ \delta(x_0-x)=\begin{cases}1 & \text{if } x_0-x>0,\\ 0 & \text{if } x_0-x\le 0\end{cases} $$ or, equivalently, $$ \delta(x_0-x)=\begin{cases}1 & \text{if } x<x_0,\\ 0 & \text{if } x\ge x_0.\end{cases} $$