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Continuous functions are of utmost importance in mathematics 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.

Contents 1 Classification of discontinuities 2 Examples 3 The set of discontinuities of a function 4 See also 5 Notes 6 References 7 External links

Consider a function f of real variable x with real values defined in a neighborhood of a point x₀. Then three situations are possible:

1. The one-sided limit from the negative direction

and the one-sided limit from the positive direction

at x₀ exist, are finite, and are equal. Then, if f(x₀) is not equal to L, x₀ is called a removable discontinuity. This discontinuity can be removed (so f can be made continuous at x₀) by setting f(x₀) = L.

2. The limits L ⁻ and L ⁺ exist and are finite, but not equal. Then, x₀ is called a jump discontinuity or step discontinuity.

3. One or both of the limits L ⁻ and L ⁺ does not exist or is infinite. Then, x₀ is called an essential discontinuity.

The term removable discontinuity is sometimes (improperly) used for cases in which the limits in both directions exist and are equal, while the function is undefined at the point x₀.^[1] This use is improper because continuity and discontinuity of a function are concepts defined only for points in the function's domain.

1. Consider the function

Then, the point x₀ = 1 is a removable discontinuity.

2. Consider the function

Then, the point x₀ = 1 is a jump discontinuity.

3. Consider the function

Then, the point x₀ = 1 is an essential discontinuity. For it to be an essential discontinuity, it would have sufficed that only one of the two one-sided limits did not exist or were infinite.

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.

• Removable singularity
• Extension by continuityhttp://en.wikipedia.org/wiki/Regular_space#Extension_by_continuity

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

• Malik, S. C.; Arora, Savita (1992). Mathematical analysis, 2nd ed. New York: Wiley. ISBN 0470218584. 

• Discontinuous on PlanetMath
• "Discontinuity" by Ed Pegg, Jr., The Wolfram Demonstrations Project, 2007.