I came across this exercise in a book that I was reading, it says:
What value would you assign to $$\int_{-\infty}^{\infty} (\delta(x))^2dx$$ ?
My attempt at a solution was to think of as so:
This integral is the same as $\int_{-\infty}^{\infty} \delta(x)\delta(x)dx$, and here I used the identity that $\int_{-\infty}^{\infty} \delta(x)f(x)dx = f(0)$.
So, I allowed that $f(x) = \delta(x)$, which then yields $$\int_{-\infty}^{\infty} (\delta(x))^2dx = \delta(0)=\infty$$
Is this a correct approach?
The product of distributions isn't really well defined, but your argument is okay.
See https://mathoverflow.net/questions/48067/is-square-of-delta-function-defined-somewhere for more information.