Let $f\colon\mathbb{R}\to\mathbb{R}$. Prove that the set $$\{x \mid \mbox{if $y$ converges to $x$, then $f(y)$ converges to $\infty$}\}$$ is countable.
My book told me to consider $g(x)=\arctan(f(x))$, then it said "it is easy to see the set is countable." But I still can't understand what it mean.