In real analysis, singularities are either discontinuities, or discontinuities of the derivative (sometimes also discontinuities of higher order derivatives). ... Borrowing from complex analysis, this is sometimes called an essential singularity.

What is the difference between singularity and discontinuity?

A point x=a is called a point/removable discontinuity if the limit of f(x) at x=a exists and is equal to L (as a two-sided limit), and f(a) is defined, but f(a) is not equal to L. A point x=a is called a removable singularity if the limit of f(x) at x=a exists, but f(a) is undefined.

What are the types of singularity?

There are basically three types of singularities (points where f(z) is not analytic) in the complex plane. An isolated singularity of a function f(z) is a point z0 such that f(z) is analytic on the punctured disc 0 < |z − z0| < r but is undefined at z = z0. We usually call isolated singularities poles.