Limit point

is an example of a function having infinitely many essential singularities with a limit point at z=0. It is an easy matter to give examples of "elementary" functions whose singularities have a countable number of limit points.

If z_0 is a limit point of S, then every ϵ-neighborhood of z_0 must contain infinitely many points of S.

As the next example shows, the set of zeros may well have a limit point not belonging to the domain.