f is constant on G

Question

Let $f$ be analytic function of complex variable that assumes only purely imaginary values in a region $G$. Prove that $f$ is constant on $G$. I am stuck here. Will anybody please help?

Answer 1

Choose any small disk $D\subseteq G$. If $f$ is constant on $D$, identity theorem implies $f$ is constant on $G$. Otherwise, by the open mapping theorem, $f(D)$ should be open subset of $\mathbb{C}$. However, it is impossible since $f(D)$ is contained in $i\mathbb{R}$. (Any open neighborhood of any point in $f(D)$ should contain other points in $\mathbb{C}\backslash i\mathbb{R}$.)