Is there a function that never vanishes, and can never reach infinity?

Question

Is there a (non-constant) function $f(z)$ in the Riemann sphere such that for all z, $f(z) \neq 0, \infty$ ?

In other words, is there a function that never vanishes, and can never reach infinity?

I am convinced there is no such function, however why is this true? Is there a magical Theorem somewhere proving this?

Answer 1

Why not $f(z) \equiv 1, \quad \forall z \in \mathbb{C}$?

Answer 2

What about $f(z)=\frac{|z|^2+1}{|z|^2+2}$?

If you want the function to be analytic, then Liouville's Theorem says that any bounded entire function must be constant.