Pierpont variation

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One of the numerical characteristics of a function of several variables which can be considered as a multi-dimensional analogue of the variation of a function of one variable. Let a function , be given on an -dimensional parallelopipedon

and let , , be a subdivision of the segment into , equal segments by points

These subdivisions generate a subdivision

of the parallelopipedon into parallelopipeda with edges parallel to the coordinate axes.


where is the oscillation of the function on (cf. Oscillation of a function). Then

If , then the function is said to be of bounded (finite) Pierpont variation on , and the class of all such functions is denoted by . This definition was suggested by J. Pierpont [1]. The class contains as a subset the class of all functions of bounded Arzelà variation on .


[1] J. Pierpont, "Lectures on the theory of functions of real variables" , 1 , Dover, reprint (1959)
[2] H. Hahn, "Reellen Funktionen" , 1 , Chelsea, reprint (1948)
How to Cite This Entry:
Pierpont variation. 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