Limit point of isolated singularities

Question

Suppose $f$ is analytic on $\mathbb{C}$ (or some open domain) except at a sequence $(c_n)$ and its limit point $c$. If each $c_n$ is a removable singularity, what can we say about $c$? While $c$ was not an isolated singularity for $f$, it becomes isolated once we remove the $c_n$'s, right? Is $c$ now necessarily a certain type of isolated singularity, or it can be either removable, pole, or essential?

What about when $a_n$'s are all poles or all essential, can we say anything about the singularity $c$? For example, is $f$ necessarily unbounded near $c$?

Answer 1

Starting from an isolated singularity $c$ of any type of a holomorphic function $f$, you can reach the first situation by just declaring $f$ not defined in the points $c_n$ of a sequence converging to $c$.

So a limit point of removable singularities can be any kind of isolated singularity after removing the singularities in the $c_n$.

If $c$ is a limit point of poles or essential singularities, every punctured neighbourhood $U$ of $c$ is a neighbbourhood of some (almost all, actually) of the poles or essential singularities, hence $f$ is unbounded on $U \setminus \{c_n\}$.