Classification of discontinuities

Continuous functions are of utmost importance in mathematics, functions and applications. However, not all functions are continuous. If a function is not continuous at a point in its domain, one says that it has a discontinuity there. The set of all points of discontinuity of a function may be a discrete set, a dense set, or even the entire domain of the function. This article describes the classification of discontinuities in the simplest case of functions of a single real variable taking real values.

The oscillation of a function at a point quantifies these discontinuities as follows:

in a removable discontinuity, the distance that the value of the function is off by is the oscillation;
in a jump discontinuity, the size of the jump is the oscillation (assuming that the value at the point lies between these limits from the two sides);
in an essential discontinuity, oscillation measures the failure of a limit to exist.

A special case is if the function diverges to infinity or minus infinity, in which case the oscillation is not defined (in the extended real numbers, this is a removable discontinuity).

Classification

For each of the following, consider a real valued function $f$ of a real variable $x$ , defined in a neighborhood of the point x₀ at which $f$ is discontinuous.

Removable discontinuity

The function in example 1, a removable discontinuity

Consider the function

f(x)={\begin{cases}x^{2}&{\mbox{ for }}x<1\\0&{\mbox{ for }}x=1\\2-x&{\mbox{ for }}x>1\end{cases}}

The point $x 0$ = 1 is a removable discontinuity. For this kind of discontinuity:

The one-sided limit from the negative direction:

L^{{-}}=\lim _{{x\to x_{0}^{{-}}}}f(x)

and the one-sided limit from the positive direction:

L^{{+}}=\lim _{{x\to x_{0}^{{+}}}}f(x)

at $x 0$ both exist, are finite, and are equal to $L$ = $L -$ = $L +$ . In other words, since the two one-sided limits exist and are equal, the limit $L$ of $f (x)$ as $x$ approaches $x 0$ exists and is equal to this same value. If the actual value of $f (x 0)$ is not equal to $L$ , then $x 0$ is called a removable discontinuity. This discontinuity can be removed to make $f$ continuous at $x 0$ , or more precisely, the function

g(x)={\begin{cases}f(x)&x\neq x_{0}\\L&x=x_{0}\end{cases}}

is continuous at $x$ = $x 0$ .

The term removable discontinuity is sometimes an abuse of terminology for cases in which the limits in both directions exist and are equal, while the function is undefined at the point $x 0$ .^[1] This use is abusive because continuity and discontinuity of a function are concepts defined only for points in the function's domain. Such a point not in the domain is properly named a removable singularity.

Jump discontinuity

The function in example 2, a jump discontinuity

Consider the function

f(x)={\begin{cases}x^{2}&{\mbox{ for }}x<1\\0&{\mbox{ for }}x=1\\2-(x-1)^{2}&{\mbox{ for }}x>1\end{cases}}

Then, the point $x 0$ = 1 is a jump discontinuity.

In this case, a single limit does not exist because the one-sided limits, $L -$ and $L +$ , exist and are finite, but are not equal: since, $L -$ ≠ $L +$ , the limit $L$ does not exist. Then, $x 0$ is called a jump discontinuity, step discontinuity, or discontinuity of the first kind. For this type of discontinuity, the function $f$ may have any value at $x 0$ .

Essential discontinuity

The function in example 3, an essential discontinuity

For an essential discontinuity, only one of the two one-sided limits needs not exist or be infinite. Consider the function

f(x)={\begin{cases}\sin {\frac {5}{x-1}}&{\mbox{ for }}x<1\\0&{\mbox{ for }}x=1\\{\frac {1}{x-1}}&{\mbox{ for }}x>1\end{cases}}

Then, the point $\scriptstyle x_0 \;=\; 1$ is an essential discontinuity.

In this case, $\scriptstyle L^{{-}}$ doesn't exist and $\scriptstyle L^{{+}}$ is infinite - thus satisfying twice the conditions of essential discontinuity. So x₀ is an essential discontinuity, infinite discontinuity, or discontinuity of the second kind. (This is distinct from the term essential singularity which is often used when studying functions of complex variables.)

The set of discontinuities of a function

The set of points at which a function is continuous is always a G_δ set. The set of discontinuities is an F_σ set.

The set of discontinuities of a monotonic function is at most countable. This is Froda's theorem.

Thomae's function is discontinuous at every rational point, but continuous at every irrational point.

The indicator function of the rationals, also known as the Dirichlet function, is discontinuous everywhere.

Notes

↑ See, for example, the last sentence in the definition given at Mathwords.

References

Malik, S.C.; Arora, Savita (1992). Mathematical Analysis (2nd ed.). New York: Wiley. ISBN 0-470-21858-4.

External links

"Discontinuous". PlanetMath.
"Discontinuity" by Ed Pegg, Jr., The Wolfram Demonstrations Project, 2007.
Weisstein, Eric W. "Discontinuity". MathWorld.
Kudryavtsev, L.D. (2001) [1994], "Discontinuity point", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4

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[1] See, for example, the last sentence in the definition given at Mathwords.