Find a complex non-constant function bounded throughout the complex plane, that is analytic in a region (not everywhere), and defined everywhere

Question

I am trying to find a non-constant function that satisfies these 3 conditions:

a) defined everywhere b) not analytic everywhere c) bounded in the complex plane meaning we can find a value $M$ such that $|f(z)| ≤ M$ for all values of $z$

I've thought of $|z|$ but that is not bounded.

Any ideas?

Answer 1

$f(z)=\begin{cases}z&\text{if }\lvert z\rvert<1\\ \sin(\operatorname{Re} z)&\text{if }\lvert z\rvert\ge 1\end{cases}$ certainly has those properties.

Answer 2

Take $$f(z)=\begin{cases}z&\lvert z\rvert \le 1\\ \frac{z}{\lvert z \rvert}&\lvert z\rvert \ge 1\end{cases}$$