Can I have any poles if $f(z) \neq 0$?
I'm trying to refresh my memory on poles... I was reading a proof for Rouche's Theorem an it, we needed to satisfy the assumptions of the argument principle theorem. At the step where I got confused, we had a meromorphic function and needed to show that it never vanished and that it had no poles on circle C. The proof proceeded to show that $f(z)\neq 0$ on C, then concluded that there were no poles and $f$ never vanished on C.
I understand how $f(z)\neq 0$ shows that $f$ never vanishes, but my question is does that also prove there are no poles?