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Discontinuous Function: Types of Discontinuity

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Contents:

  1. What is a Discontinuous Function?
  2. Graph of a Discontinuous Function
  3. Finding Discontinuities
  4. Types of Discontinuity

What is a Discontinuous Function?

A discontinuous function is a function which is not continuous at one or more points. Most functions are, perhaps surprisingly, discontinuous in one way or another [1].


Being “continuous at every point” means that at every point a:

  1. The function exists at that point. If you can plug an x-value into your function and it returns a value, it’s continuous at that point.
  2. The limit of the function as x goes to the point a exists. In other words, the function values surrounding point “a” are all approaching the same number.
  3. Both (1) and (2) are equal.

In notation, we can write that as:
discontinuous function

In plain English, what that means is that the function passes through every point, and each point is close to the next: there are no drastic jumps (unlike jump discontinuities). When you’re drawing the graph, you can draw the function from left to right without taking your pencil off the paper.

A discontinuous function is one for which you must take the pencil off the paper at least once while drawing.

Graph of a Discontinuous Function

Graphically, a discontinuous function will either have a hole—one spot, or several spots, where the function is not defined—or a jump, where the value of f(x) changes (“jumps”) quickly as you go from one spot to another that is infinitesimally close.

what is a discontinuous function

A vertical asymptote. The function will approach this line, but never actually touch it.

It might also have an asymptote, a line where, as the function approaches, it goes to infinity. The function never merges with this line, though it may approach infinitely close.

Finding Discontinuities

If your function can be written as rational function (i.e. a fraction), any values of x that make the denominator go to zero will be discontinuities of your function. Those are the places your function is not defined because of division by zero.

If you have a piecewise function, the point where one piece ends and another piece ends are also good places to check for discontinuity.

Otherwise, the easiest way to find discontinuities in your function is to graph it. Take note of any holes, asymptotes, or jumps. These all represent discontinuities, and just one discontinuity is enough to make your function a discontinuous function.

Types of Discontinuity

types of discontinuity

Types of Discontinuity.



The four main types of discontinuity are:

  1. Removable,
  2. Jump (Step),
  3. Infinite,
  4. Oscillating,

1. Removable

hole in function removable discontinuous

A removable discontinuity (a hole in the graph).


Removable discontinuities are where the limits are equal (it’s a hole, not a jump). It’s called removable because the hole can be filled in.

2. Jump

Jump (or Step) discontinuities are where there is a jump or step in a graph. The following graph jumps at the origin (x = 0).
non continuous function

3.Infinite

graph of 1/x

Graph of y = 1/x, which tends towards both negative and positive infinity at x = 0.


Infinite discontinuities are when the limit at the gap tends towards infinity. This can be as the function approaches the gap from either the left or the right. This type of discontinuity is normally associated with having a vertical asymptote [2].

4. Oscillating

Oscillating discontinuities jump about wildly as they approach the gap in the function. They are sometimes classified as sub-types of essential discontinuities.

oscillating discontinuity

It’s impossible to pin down a limit for the function sin(1/x) at zero. It has an oscillating discontinuity at that point.


Discontinuous Function: A Note on Classifying Types of Discontinuity

Classifying types of discontinuity is more difficult than it appears, due to the fact that different authors classify them in different ways. For example:

  • Some authors simplify the types into two umbrella terms: removable (holes) and non-removable (jumps, infinite and essential discontinuities cannot be removed as they are too far apart or wild in their behavior).
  • Essential discontinuities (that jump about wildly as the function approaches the limit) are sometimes referred to as the “non-removable discontinuity”, excluding jumps and infinite from the definition of non-removable.
  • Some authors also include “mixed” discontinuities as a type of discontinuity, where the discontinuity is a combination of more than one type.

The takeaway: There isn’t “one” classification system for types of discontinuity that everyone agrees upon. Which system you use will depend upon the text you are using and the preferences of your instructor.

References

[1] Drago et. al. A “Bouquet” of Discontinuous Functions for Beginners in Mathematical Analysis. Retrieved July 13, 2021 from: http://ceadserv1.nku.edu/longa//classes/mat420/days/highlights/PathologicalFunctions.pdf
[2] Section 1.4. Continuity. Retrieved July 13, 2021 from: https://www.math.uh.edu/~beatrice/143114.pdf


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Stephanie Glen. "Discontinuous Function: Types of Discontinuity" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/discontinuous-function/
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