We know that, if $p : \mathbb C \to \mathbb C$ is a nontrivial polynomial, then the function $P: \mathbb C P^1 \to \mathbb C P^1$ defined as $P([z:1]) = [p(z):1]$ and $P([1:0]) = [1:0]$ is a smooth map; this can be checked in a straightforward manner.
I am wondering if this can be generalized to a general proper smooth map $f: \mathbb C \to C$, i.e. whether
(1) The function $F: \mathbb C P^1 \to \mathbb C P^1$ defined as $F([z:1]) = [f(z):1]$ and $F([1:0]) = [1:0]$ is continuous, and
(2) Whether the function $F$ thus defined is smooth.
I am told that the answer for (1) is true while the answer for (2) is no. However, I am having a hard time proving this. It would be great if you could provide a counterexample to (2).
EDIT: I made an important edit, that the map $f$ is smooth proper map.