# Non analytic, non constant , continuous complex valued function

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

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

• i want to know the points where f is not analytic except zero Mar 28, 2020 at 8:35
• @PrashantDattatrey Well, check for yourself. for $h\in\mathbb{R}^+$, compute $[f(z+h)-f(z)]/h$ and take the limit as $h\to0^+$. Compute also $[f(z+ih)-f(z)]/(ih)$ and take the limit as$h\to0^+$. Are these two equal? Mar 28, 2020 at 8:43

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 $$\mathbb C/{0}$$.

• $f(z) = arg (z)$ is not continuous on $\Bbb C\setminus0$ unless you delete a ray (originating from origin) from complex plane Mar 28, 2020 at 17:09
• You are right. I fixed that now. Mar 28, 2020 at 17:23
• This Arg (Z) should be principal argument Mar 28, 2020 at 17:32
• yes it is. There is big A. Mar 28, 2020 at 17:33