Holomorphic funtion with $f(n) = n$ for all $n \in \mathbb N$

Question

I really struggle with this seemingly harmless question:

Let $f: \mathbb C \to \mathbb C$ be holomorphic with $f(n) = n$ for all $ n \in \mathbb N$. Does this imply $f(z) = z$ for all $z \in \mathbb C$?

I don't really know what's the right way to solve this. Obviously we can't use the Identity theorem, since $n \to \infty $. I doubt that I explicitly have to construct a function rather than giving an easy counter example. Any hints?

Answer 1

Can you think of a nonzero function that vanishes at the integers?

Hint: think trig.