# Question on divisors of meromorphic functions on Riemann surfaces

I have a question on the following definition

I don't quite get why the definition says " if f is identically zero in a neighborhood of a", I mean there's nothing wrong with that I just don't get why it's not saying " if f vanishes everywhere" since $$f$$ would be constant zero if it would vanish on an open set due to the identity theorem, right?