Let $U\subseteq \mathbb{C}$ be a region and let $h,g$ meromorphic functions in $U$. Suppose that there exist a set $W\subseteq U$ with a acumulation point in $U$ with $h(w)=g(w)$ for all $w\in W$.
If $h$ has an essential singularity in $U$. Can I say that $h=g$?
Remark: The case that $h$ has no essential singularities was solved in the comments of the post:'Identity theorem' for Meromorphic functions.
Therefore, I think that the important question is what happens if $W$ has the limit point in an essential singularity of $h$ in $U$.