Proposition

Prove that if $f$ is entire and $\Im(f)$ is constant on the closed unit disk, then $f$ is constant.

Solution

Let $c$ be the value that $f$ takes on the closed unit disk. Let $g(z) = c$ for all $z$. $g$ is clearly entire. Then $f$ and $g$ agree on the closed unit disk, which has an accumulation point in $\mathbb{C}$. Then $f = g$ on $\mathbb{C}$ by the identity theorem.