Non analytic, non constant , continuous complex valued function

Question

I am trying to construct a non analytic , non constant, continuous function $f $ such that $f$ :$ \mathbb C\setminus\{0\}\to \mathbb C$ with $f(z)= f\left(\frac{z}{|z|}\right)$. I got stuck how to start this problem

Answer 1

How about $f(z)=z/|z|$? This is continuous on $\mathbb{C}-\{0\}$, not analytic since $|z|$ is involved, non constant and we have $f(z/|z|)=(z/|z|)/|z/|z||=(z/|z|)/1=z/|z|=f(z)$

Answer 2

Note that the condition of non-constant is trivial for any non-analytic $f$.

Using polar coordinates $z=r.e^{i\theta}$

$f(r.e^{i\theta})=f(\frac{r.e^{i\theta}}{|r.e^{i\theta}|})$

$f(r.e^{i\theta})=f(e^{i\theta})$

A convenient choice will be $f(r.e^{i\theta})=\theta^2\implies f(z)=(Arg(z))^2$ which is defined on $\mathbb C/{0}$.