# Function of bounded variation

A function whose variation is bounded (see Variation of a function). The concept of a function of bounded variation was introduced by C. Jordan [1] for functions of one real variable in connection with a generalization of the Dirichlet theorem on the convergence of Fourier series of piecewise-monotone functions (see Jordan criterion for the convergence of a Fourier series). A function defined on an interval is a function of bounded variation if and only if it can be represented in the form , where (resp., ) is an increasing (resp., decreasing) function on (see Jordan decomposition of a function of bounded variation). Every function of bounded variation is bounded and can have at most countably many points of discontinuity, all of the first kind. A function of bounded variation can be represented as the sum of an absolutely-continuous function (see Absolute continuity), a singular function and a jump function (see Lebesgue decomposition of a function of bounded variation).

In the case of several variables there is no unique concept of a function of bounded variation; there are several definitions of the variation of a function in this case (see Arzelà variation; Vitali variation; Pierpont variation; Tonelli plane variation; Fréchet variation; Hardy variation).

#### References

[1] | C. Jordan, "Sur la série de Fourier" C.R. Acad. Sci. Paris , 92 (1881) pp. 228–230 |

[2] | I.P. Natanson, "Theorie der Funktionen einer reellen Veränderlichen" , H. Deutsch , Frankfurt a.M. (1961) (Translated from Russian) |

**How to Cite This Entry:**

Function of bounded variation.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Function_of_bounded_variation&oldid=11751