Proof that all rational maps $\mathbb{P}^1\to\mathbb{P}^N$ are regular without using codimension?
Solution 1:
Suppose $f:P^1\to P^n$ is a rational map. On a nonempty open subset of the domain, then, it can be written as $(x:y)\mapsto(f_0(x,y):\cdots:f_n(x,y))$ with the $f_i$ homogeneous of the same degree.
If all the $f_i$ vanish at a point, they all have a factor in common, and you can remove it. That way, you can extend the map to the whole of $P^1$.