Identity theorem - entire function

Question

Let $f$ an analytic function on $\mathbb{C}$ such that $f(e^{it})=e^{it}$ for all $t \in \mathbb{R}$. Compute $f(z)$ for all $z \in \mathbb{C}$.

Is anyone could give me a hint how to solve the problem?

Answer 1

Easier: two analytic functions on $\mathbb{C}$ agreeing on a set with a limit point must be equal. This function agrees with $z\mapsto z$ on the unit circle.