Let $\delta(x)$ be a Dirac-delta function. I am wondering how to prove the following:
\begin{align}
\delta(ax, by) = \frac{1}{|ab|} \delta(x,y)
\end{align}
Because of the definition (if I understand it correctly),
\begin{align}
\delta(ax, by) = \begin{cases}
\infty & \text{if $ax = 0,\ by=0,$}\\
0 & \text{otherwise.}
\end{cases}
\end{align}
If this is true, the scale of the argument should not influence the value of the delta function itself? Correct me if I'm wrong.