Holomorphic Function with Constant Imaginary Part is Constant/Corollary

Theorem

Let $D \subseteq \C$ be an open subset of the set of complex numbers $\C$.

Let $f: D \to \C$ be a holomorphic function such that $\map \Re f$ is constant.

Then $f$ is constant.

Proof

From Combination Theorem for Complex Derivatives: Multiple Rule, $i f$ is holomorphic.

From Imaginary Part of Imaginary Unit times Element of *-Algebra, we have:

$\map \Im {i f} = \map \Re f$

Hence $\map \Im {i f}$ is constant.

From Holomorphic Function with Constant Imaginary Part is Constant, there exists $c \in \C$ such that:

$i \map f z = c$

for all $z \in D$.

Hence $\map f z = -i c$ for all $z \in D$.

So $f$ is constant.

$\blacksquare$