I'm a bit confuse with something, when we say that $$\Delta f(x)=\delta _{0},$$ I know that it mean that for all $\varphi \in \mathcal C_0^\infty (\mathbb R)$, $$\int_{\mathbb R}\varphi \Delta f=\varphi (0).$$
I'm not really sure how to interpret this. Is the idea behind is $\Delta f(x)=0$ if $x\neq 0$ and $\Delta f(0)=\infty $ ? And thus $\Delta f(x)=0$ p.p. or absolutely not ?