# Weak solutions for divisors

I have a question on the following definition in the Forster:

I don't get the part where it says "Clearly a weak wolution $$f$$ is a proper, i.e., meromorphic function, solution precisely if $$f$$ is holomorphic on $$X_D$$".

First, I don't get why he says "proper i.e. meromorphic", proper and meromorphic functions aren't the same, are they?

Also I would say $$f$$ is then only a solution on $$X_D$$ not on the whole of $$X$$. Why is $$f$$ then a solution on all of $$X$$?