While continuous functions are important in mathematics, not all functions are continuous. If a function is not continuous at a limit point (also called an "accumulation point" or "cluster point") of its domain, 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.

In elementary real analysis, discontinuities of real functions of one real variable are often distinguished according to the behavior of one-sided limits. While a classification is not entirely standard, a common division is between discontinuities of the first kind, where the relevant one-sided limits exist, and discontinuities of the second kind, where at least one one-sided limit fails to exist or is infinite. Special cases include removable discontinuities and jump discontinuities.

Classification

For each of the following, consider a real valued function is discontinuous.

Removable discontinuity

Consider the piecewise

is a removable discontinuity. For this kind of discontinuity:

The one-sided limit both is not is called a

The term removable discontinuity is sometimes broadened to include a removable singularity, in which the limits in both directions exist and are equal, while the function is undefined This use is an abuse of terminology because continuity and discontinuity of a function are concepts defined only for points in the function's domain.

Jump discontinuity

is a .

exist and are finite, but are not is called a jump discontinuity, step discontinuity, or discontinuity of the first kind

Essential discontinuity

).

is an .

is an essential discontinuity, infinite discontinuity, or discontinuity of the second kind. (This is distinct from an essential singularity, which is often used when studying functions of complex variables).

Counting discontinuities of a function

has a removable has a jump has an essential

are relevant in the literature.

• is an F_{\sigma} set. The set of points at which a function is continuous is always a G_{\delta} set (see).

• is at most countable This is Froda's theorem.

Tom Apostol follows partially the classification above by considering only removable and jump discontinuities. His objective is to study the discontinuities of monotone functions, mainly to prove Froda’s theorem. With the same purpose, Walter Rudin and Karl R. Stromberg is always a countable set (see).

The term essential discontinuity has evidence of use in mathematical context as early as 1889. However, the earliest use of the term alongside a mathematical definition seems to have been given in the work by John Klippert. into the three following sets:

is called an essential discontinuity of first kind is said an essential discontinuity of second kind. without losing its characteristic of being countable, by stating the following:

• is countable.

Rewriting Lebesgue's theorem

is a bounded function in the regard of the Riemann integrability In fact, Lebesgue's theorem (also named Lebesgue-Vitali theorem) is a set with Lebesgue's measure zero.

The main discontinuities for that purpose are the essential discontinuities of first kind and consequently the Lebesgue-Vitali theorem can be rewritten as follows:

• if and only if has Lebesgue's measure zero.

:

• (see)

• is a regulated function is Riemann integrable

Examples

Thomae's function is discontinuous at every non-zero rational point, but continuous at every irrational point. One easily sees that those discontinuities are all removable. By the first paragraph, there does not exist a function that is continuous at every rational point, but discontinuous at every irrational point.

The indicator function of the rationals, also known as the Dirichlet function, is discontinuous everywhere. These discontinuities are all essential of the first kind too.

Consider now the ternary Cantor set One way to construct

used in the \mathcal{C} (In another way, the same conclusion follows taking into account that \mathcal{C} is a closed set

is an uncountable set with null Lebesgue measure is a null Lebesgue measure set and so in the regard of Lebesgue-Vitali theorem is a Riemann integrable function.

then no neighbourhood

Discontinuities of derivatives

. According to Darboux's theorem in which case Bolzano's theorem

. This means in particular that the following two situations cannot occur:

Furthermore, two other situations have to be excluded (see John Klippert): fails to possess an antiderivative.

, is said to be a fundamental essential discontinuity if

.

is a fundamental one.

See also

Notes

References

Sources

External links

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