Variation of a function

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A numerical characteristic of functions of one real variable which is connected with differentiability properties.

1) Let be a complex-valued function defined on an interval ; its variation is the least upper bound of sums of the type

where is an arbitrary system of points on . This definition was given by C. Jordan [1]. If , one says that has (is of) bounded (finite) variation over , and the class of all such functions is denoted by . A real-valued function belongs to the class if and only if it can be represented in the form , where and are functions which increase on (the Jordan decomposition of functions of bounded variation). The sum, the difference and the product of two functions of class are also functions of class . This is also true of the quotient of two functions of class if the modulus of the denominator is larger than a positive constant on . Every function in is bounded and cannot have more than a countable set of discontinuity points, all of which are of the first kind. All these properties of functions in were established by Jordan [1] (see also [2]).

Functions in are almost-everywhere differentiable on and may be represented as

where is an absolutely continuous function, is a singular function and is a saltus function (the Lebesgue decomposition of a function of bounded variation). Such a decomposition is unique if [3], [2].

The class was originally introduced by Jordan in the context of the generalization of the Dirichlet criterion for the convergence of Fourier series of piecewise-monotone functions. It was shown by him that Fourier series of -periodic functions in the class converge at all points of the real axis. Functions of bounded variation subsequently found extensive application in various branches of mathematics, especially in the theory of the Stieltjes integral.

One sometimes also considers classes , defined as follows. Let (, ) be a continuous function which increases monotonically if . Let be the least upper bound of sums of the type

where is an arbitrary system of points in . The quantity is called the -variation of on . If , one says that has bounded -variation on , while the class of such functions is denoted by ([4]). If , one obtains Jordan's class , while if , , one obtains Wiener's classes [5]. The definition of the class was proposed by L.C. Young [6].



In particular, on any interval ,

for , , these being proper inclusions.


[1] C. Jordan, "Sur la série de Fourier" C.R. Acad. Sci. Paris Sér. I Math. , 92 : 5 (1881) pp. 228–230
[2] I.P. Natanson, "Theory of functions of a real variable" , 1–2 , F. Ungar (1955–1961) (Translated from Russian)
[3] H. Lebesgue, "Leçons sur l'intégration et la récherche des fonctions primitives" , Gauthier-Villars (1928)
[4] N.K. [N.K. Bari] Bary, "A treatise on trigonometric series" , Pergamon (1964) (Translated from Russian)
[5] N. Wiener, "The quadratic variation of a function and its Fourier coefficients" J. Math. and Phys. , 3 (1924) pp. 72–94
[6] L.C. Young, "Sur une généralisation de la notion de variation de puissance borneé au sens de M. Wiener, et sur la convergence des series de Fourier" C.R. Acad. Sci. Paris Sér. I Math. , 204 (1937) pp. 470–472


The variation of a function as defined above is often called the total variation. It is the sum of the negative and positive variations (cf. Negative variation of a function; Positive variation of a function). One has

where is the Banach indicatrix of . If , then


[a1] E. Hewitt, K.R. Stromberg, "Real and abstract analysis" , Springer (1965) pp. 266; 270; 272
[a2] S. Saks, "Theory of the integral" , Hafner (1937) (Translated from French)

Several different definitions of variation exist for functions of several variables (Arzelà variation; Vitali variation; Pierpont variation; Tonelli plane variation; Fréchet variation; Hardy variation). The following definition, [1], based on the use of the Banach indicatrix, also proved very fruitful. Let a real-valued function be given and be Lebesgue-measurable on an -dimensional cube . The variation of order , where , of on is the number

where denotes the -th variation of the set (cf. Variation of a set), while the integral is understood in the sense of Lebesgue. This definition allows one to transfer many properties of functions of bounded variation in one variable to functions of several variables. For instance,

a) ;

b) If a sequence of functions , converges uniformly to in , then

c) If the function is continuous in and all its variations are finite, has a total differential almost-everywhere.

d) If the function is absolutely continuous in , then

e) If the function is continuous in a cube with side-length , if it has bounded variations of all orders in and if it can be periodically extended with period for all arguments , , in the -dimensional space, then its Fourier series converges uniformly to it on (Pringsheim's theorem).

A sufficient condition for being of bounded variation is: If the function has continuous derivatives of all orders up to and including in the cube , then its variation of order is finite. This theorem is a final theorem in the sense that the smoothness conditions cannot be improved for any .


[1] A.G. Vitushkin, "On higher-dimensional variations" , Moscow (1955) (In Russian)

A.G. Vitushkin

How to Cite This Entry:
Variation of a function. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by B.I. Golubov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article