I'm currently following A. Gathmann's Algebraic Geometry, chapter 5 - varieties. I've seen the concept of prevarieties (ringed spaces with finite covers of affine varieties) and I'm stuck with the concept of gluing of prevarieties. I've understood the compatibility conditions on the construction, but when dealing with morphisms $f\colon X \to Y$ where $X$ and $Y$ are prevarieties, I don't know how to express the conditions of continuity and regularity of the function.
To be explicit: in exercise 5.7a), the author asks for a proof that every morphism $f \colon \mathbb A^1\setminus \{ 0 \} \to \mathbb P^1$ extends to the whole line. I solved that exercise using the usual homogeneous coordinates for $\mathbb P^1$ - pick the usual affine opens, show that the regular function is a quotient of 1-variable polynomials, clear out denominators etc. However, I can't even write down what it means for $f$ to be continuous or regular when taking $\mathbb P^1$ as the glued prevariety, i.e. a quotient of $\mathbb A^1 \sqcup \mathbb A^1$ modulo $x \to \frac{1}{x}$ on the opens $\mathbb A^1\setminus \{0\}$. This question seems to address my desired approach to this problem, but my doubts are more elementary.
As a matter of showing effort, I tried characterizing all of $\mathbb P^1$ open sets using the Zariski (cofinite) topologies that came from $\mathbb A^1$, but it seems too complicated. That does also say nothing about $f$. Since I was stuck in continuity, regularity was a worse matter because (I think) I have to consider things like $f^* \varphi$ where $\varphi \in \mathcal O_{\mathbb P^1}(U)$, and or that I need the continuity of $f$.
Any help is appreciated.