Problem: Consider a sequence of functions $(f_n)_n$ from $\mathbb{R}$ to $\mathbb{R}$ which converges pointwise to some function $f:\mathbb{R} \to \mathbb{R}$. Furthermore, we assume that the convergence is uniform on $\mathbb{R}_0 =\mathbb{R}\setminus \{0\}$. Can you conclude that $(f_n)_n$ converges uniformly on whole $\mathbb{R}$?
Attempt: I suggest this will be true? But I don't really know how start. First I tried something in the fashion of: for all $x\neq 0$ we have $$|f_n(0)-f(0)| \le |f_n(0)-f_n(x)| +|f_n(x) -f(x)| +|f(x)-f(0)|$$ and then making each of these terms small enough. However, this won't work for the first and last term where continuity would be required. So this strategy will fail? A second attempt I had in mind was proving that we can somehow extent the uniform convergence of $\mathbb{R}_0$ to its closure $\mathbb{R}$. But I don't know how to start?
Given $\varepsilon>0$. You know $\lvert f_n(0)-f(0)\rvert<\varepsilon$ for all $n>N_0$ from pointwise convergence, and you know for all $x\in\mathbb{R}_0$ $\lvert f_n(x)-f(x)\rvert<\varepsilon$ for all $n>N_1$ from uniform convergence on $\mathbb{R}_0$. Can you combine the two bounds $n>N_0$ and $n>N_1$?