I already know that $\delta^{2}(t)$ is not defined.
However consider $A(t) = \delta(t) \cdot \delta(t-t_0)\ $ with $ t_0 \neq 0 $. Can I claim that $A(t) = 0$ ? If that is the case, is there a way to prove it besides just using the non rigorous argument that $\delta(t) = 0$, $\forall t \neq 0$ and $\delta(0) = \infty$. Even then we would have $@t = 0$ for example $0 \cdot \infty$ which is indeterminate.