How are zeros of a global analytic function defined? Are they isolated?

Question

According to Wikipedia,

An analytic function in an open set U is called a function element.

A global analytic function is a family f of function elements such that, for any (f,U) and (g,V) belonging to f, there is a chain of analytic continuations in f beginning at (f,U) and finishing at (g,V).

In this regard, I have the following questions:

1. What is meant by saying that a global analytic function vanishes on an open set in the complex plane?

2. If the above question makes sense, then does it imply that the global analytic function is identically zero?

Some references for the above would be very helpful.

Thanks in advance.