How to show that a function is not a regular map?

Question

I'm proving that if $f:\mathbb{C}\to\mathbb{C}$ is a bijective map that is not a polynomial. So, $f$ is continuous in the Zariski topology but that it is not a regular map. I have proven that it is continuous but I do not know if I am proving the second part correctly:

$\textbf{My attempt:}$ If I assume that $f$ is a regular function, then I know that exist $f\in\mathbb{C}[x]$ such that $f$ is the component function of $f$, but this implies that $f$ is a polynomial function, but is not possible by hypothesis.

Answer 1

I think you're right, the main part of the statement is proving that $f$ is continuous in the Zariski topology.