To find analytic function with given condition How to find  all analytic function on the disc $\{z:|z-1|<1 \}$ with $f(1)=1$ and $f(z)=f(z^2)$ ?.

For any $z = re^{i\theta}, r \gt 0$ in the disc, the sequence $z_n = r^{1 \over 2^n} e^{i \theta /2^n}$ will converge to $1$, and $f(z_n) = f(z_n^{2^n}) = f(z)$.  Since $f$ is continuous, $f(z) = \lim_{n \rightarrow \infty} f(z_n) = f(\lim_{n \rightarrow \infty} z_n) = f(1) = 1$
