Complex analysis , $f$ : $C$ $\to$ $C$ , $f$ is continuous and $f(z) = f(2z)$ prove that $f(z)$ = $\alpha$

Question

If $f$ : $C$ $\to$ $C$

$f$ is continuous and $f(z) = f(2z)$

prove that $f(z)$ = $\alpha$ , where $\alpha$ is constant .

I'm trying to use the definition of continuity

$\lim_{z\to w}$ $f(z) = f(w)$

to prove it but i dont know how to use that $f(z) = f(2z)$ in this proof ,

any hints please

Answer 1

Hint: Turn $f(z) = f(2z)$ around and note that $f(z) = f(z/2) = f(z/4) = \cdots$. Now use that $f$ is continuous at the origin to reach the conclusion.