I'm learning about the space of germs of holomorphic functions. As far as I understand we define the space $\mathcal{O}_x$ to be the set of "germs" i.e. equivalence classes of functions that coincide on a certian neighbourhood of the point $x$.
I haven't seen this addressed anywhere so this may be a silly question, but doesn't the identity principle imply that each such germ contains only one function, in any meaningful way? I suppose you could of course define two functions $f$, $g$ so that they are both the same but one has a smaller domain, or they have disconnected domains and differ on some different components.
So i suppose my question is this: It seems like the identity principle implies that two holomorphic functions have the same germ only if they are "essentially the same", i.e. can be extended to the same function on some connected domain? Am I wrong in saying this? And if not, doesnt this make the distinction between holomorphic germs and functions a bit pointless, since each of these equivalence classes would only really contain one function?