Let $g$ be nonconstant and entire, give the relation between the sets $\operatorname{cl} \{ z \in \mathbb{C} : |g(z)| > 1 \}$ and $\{ z \in \mathbb{C} : |g(z)| = 1 \}$.
I'm probably missing something big here as all I got is that when we replace $g$ by an infinitely differentiable $f:\mathbb{R} \rightarrow \mathbb{R}$, the latter is not a subset of the former.