Theorems in complex analysis. [closed]
I need help to prove these theorems: 1)If $\mathcal f $ is continuous on $\mathbb C $ and $\mathcal f(z) = f(2z) \forall z \in \mathbb C $ then $\mathcal f $ is constant . 2)If $\mathcal f $ is entire and $\mathcal f(2z)=2f(z)$ then $ \exists $ $\lambda $ such that $\mathcal f(z)=\lambda\mathcal z $.
Hint: $f(z/2^n) =f(z)$ and $\lim f(z/2^n) =f(0)$