A regular point for a complex-valued function is where the function is analytic, while a singular point is where it is not, but at least one nearby point is analytic. An isolated singular point is one where the function is analytic in a deleted neighborhood around it, whereas a non-isolated singularity has singularities in every deleted neighborhood. Examples include the function 1/z, which has an isolated singularity at z = 0 and at every integer point.

1. Singularities

3. A point z = a is called a regular point for
a complex-valued function f if f is analytic at a.
Point a is called a singular point or a
singularity, of f, if f is not analytic at a but
every neighborhood of a contains at least one
point at which f is analytic.
Singularity of f:

5. A singular point a is said to be an isolated
singular point or an isolated singularity of f
if f is analytic in some deleted neighborhood
of a. Otherwise, we say a is a non-isolated
singular point of f.
Equivalently, we say that a point a is a
non-isolated singularity of f if a is a
singularity and every deleted neighborhood
of a contains at least one singularity of f .
Isolated singularity

6. Example,
1. 1/z has an isolated singular
point at z = 0
2. has isolated
singularities at every integer point