**Contents:**

Essential Discontinuity

Essential Singularity

## Essential Discontinuity

An **essential discontinuity** (also called *second type* or *irremovable* discontinuity) is a discontinuity that jumps wildly as it gets closer to the limit. This makes it difficult to remove the gap (hence the alternate name, “irremovable” discontinuity) and perform any calculations on the function.

An essential discontinuity is considered to be the ‘worst kind’ of discontinuity. That’s because the behavior around where the limits *should* be is** abnormal, impossible to calculate, and sometimes just plain crazy.** There might be many jump discontinuities within a very short distance, or you might not be able to pin down any kind of behavior at all. Graphing calculators might not be any help (because of the aberrant behavior), and you might have to resort to pen and paper to make any sense of the graph.

## What is a Discontinuity?

A

*discontinuity*is a point where a function is not continuous. Several different types of discontinuity exist; An essential discontinuity is where the behavior of the function, as x approaches the discontinuity, is aberrant. More specifically—one or more of the one-sided limits don’t exist.

## Subtypes (Oscillatory and Infinite Discontinuity)

Discontinuities are broken into **two subtypes**, based on whether the “gap” can be removed (i.e. filled in) or not. Discontinuities that can’t be removed are called **irremovable**, or essential; if the gap can be filled in, that’s a removable discontinuity.

Essential discontinuities (i.e. non-removable ones) can be further broken down into two types, based on whether the one-sided limits are bounded or unbounded (Bauldry, 2011):

**Bounded**:*oscillatory discontinuity*. The pattern near the limit bounces up and down, never forming a pattern you can pin down.**Unbounded**:*infinite discontinuity*. The limits exist, but they are infinite, getting larger as you move closer to the limit.

Simple (removable) discontinuities can also be broken down into two subtypes:

- A
**removable discontinuity**has a gap that can easily be filled in, because the limit is the same on both sides. - A
**jump discontinuity**at a point has limits that exist, but it’s different on both sides of the gap.

In either of these two cases the limit can be quantified and the gap can be removed; An essential discontinuity *can’t *be quantified. Note that **jump discontinuities that happen on a curve** can’t be removed, and are therefore essential (Rohde, 2012).

## Essential Singularity

An **essential singularity** is an ill-behaved “hole” in a non-analytic complex function that can’t be removed / repaired. In other words, there’s no easy way to turn a function with an essential singularity into one that’s continuous and differentiable.

This type of singularity is similar to its real-valued counterpart: the essential discontinuity. These types of singularities / discontinuities are difficult to deal with because of their **pathological behavior** at a certain point.

Essential singularities are one of **three types of singularity** in complex analysis. The other two are poles (isolated singularities) and removable singularities, both of which are relatively well behaved. Essential singularities are **classified by exclusion:** if it isn’t a pole or a removable singularity, then it’s an essential one.

## Example of a Function with an Essential Singularity

The function exp (1/z) has an essential singularity at z = 0, where the function is undefined (because of division by zero). At this point, the function does not have a limit, so it’s impossible to remove the singularity.

## In Terms of the Laurent Series

Essential discontinuities can be identified by looking at the behavior of a Laurent series representing the neighborhood around a singularity. Specifically, a singularity is essential if the principal part of the Laurent series has infinitely many nonzero terms (Kramer, n.d.).

## References

Bauldry, W. (2011). Introduction to Real Analysis: An Educational Approach. John Wiley & Sons.

Knopp, K. “Essential and Non-Essential Singularities or Poles.” §31 in Theory of Functions Parts I and II, Two Volumes Bound as One, Part I. New York: Dover, pp. 123-126, 1996.

Kramer, P. L.S. Examples. Retrieved August 22, 2020 from: http://eaton.math.rpi.edu/faculty/Kramer/CA13/canotes111113.pdf

Krantz, S. G. “Removable Singularities, Poles, and Essential Singularities.” §4.1.4 in Handbook of Complex Variables. Boston, MA: Birkhäuser, p. 42, 1999.

Rohde,U. et al. (2012). Introduction to Differential Calculus: Systematic Studies with Engineering Applications for Beginners. John Wiley and Sons.

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