Let $f$ be a holomorphic function on $\mathbb{C}\setminus A$, where $A$ is a set of isolated singularities. I know it is possible, that $A$ contains infinite (countable) number of poles, but:
1) Is it possible that $A$ contains infinite (countable) number of essential singularities?
2) Is it possible to construct a function that has exactly one pole of every order $n\in\mathbb{N}$?
3) If both $f$ and $g$ have essential singularities at some $z\in \mathbb{C}$, is it possible that $f+g$ does not have essential singularity there? (I think it's easy, just define $g\equiv-f$)
4) Does identity theorem hold for meromorphic functions? What is counterexample if not?
Thanks for help