Why this application is well-defined?

Question

Let $C$ be a curve. An application $\phi:C\to \mathbb P^n$ is called regular in a point $P\in C$ if there are regular functions $f_0,\ldots,f_n$ defined in a neighborhood $V$ of $P$ in $C$ such that

$$\phi(x)=[f_0:\ldots:f_n]$$

And there is some $f_i$, such that $f_i(P)\neq 0$.

My question is why this application is well-defined.

More specifically:

1.Why $x_1=x_2\implies \phi(x_1)=\phi(x_2)$?

2.Why if $f_i(P)\neq 0$, then there is some neighborhood of $P$ which $f_i$ is not zero?

Thanks in advance

Answer 1

1. This is injectivity, which we don't want (to assume). $\phi$ is a map by assumption.
2. Well, $D(f_i) = \{Q : f_i(Q) \neq 0\}$ is such a neighborhood.