is the real part of a holomorphic function holomorphic?

Question

Say we have some entire function $f:\mathbb{C} \rightarrow \mathbb{C}$. Does this guarantee that the function $Re(f)$ will also be entire?

Answer 1

The only entire holomorphic functions whose imaginary parts are constant are the constant functions. So $\operatorname{Re}(f)$ is holomorphic if and only if it is constant (which implies that $f$ itself is constant).

Answer 2

If $\mathfrak{R}(f)$ is entire, then $\exp(i\mathfrak{R}(f))$ is also entire. But $|\exp(i \mathfrak{R}(f))| \leq 1$, so $\exp(i\mathfrak{R}(f))$ is constant (Liouville's theorem) and thus so is $\mathfrak{R}(f)$. Hence, $f$ has constant real part, and the C-R equations get you that $f$ has constant imaginary part as well.