Let $C$ be a smooth and irreducible projective curve, and let $f: X \rightarrow \mathbb{P}^1(\mathbb{C})$ a morphism of varieties: then $f$ is either constant or surjective. I am trying to prove this in an elementary way (I don't know about schemes, and I cannot use the notion of completeness).
I think it should be possible to cover $\mathbb{P}^1$ with two affine open sets $U,V$ both isomorphic to $\mathbb{A}^1$ and then exploit the fact that the functions from $X$ to $\mathbb{A}^1$ have to be constant but I cannot seem how to patch this up and how it relates to surjectivity.