Consider some constant function $f(x)=c$, $f(x_0)=0$, for $c\neq 0$. This function is obviously discontinuous as $x_0$, so according to the topological definition of continuity, there must exist an open set $U\subset \mathbb{R}$ such that $f^{-1}(U)$ is not open. I'm having trouble finding such a set.
Naturally, I would want to consider the interval $(c-\epsilon, c+\epsilon)$ for some $\epsilon
Thanks so much!