We know that if $f:\mathbb{R} \to \mathbb{R}$ is continuous, then its mollification $f_\epsilon$ converges uniformly to $f$ on compact subsets of $\mathbb{R}$ as $\epsilon \to 0$.
My question is, suppose $f$ is continuous everywhere except at $0$ (it blows up to infinity).
Is it possible to find a sequence via mollifcation $f_\epsilon$ such that $f_\epsilon$ converges to $f$ uniformly in the compact subsets of $\mathbb{R}$ that don't contain $0$?
Mollification with $\phi_n\in C^\infty_0$ functions will do that, since the support will shrink with increasing $n$. Hence the domain of influence of a mollifier at a certain point different from the origin will be contained in a compact set not containing $0$.