# 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.

Let 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 . The class contains as a subset the class of all functions of bounded Arzelà variation on .

How to Cite This Entry:
Pierpont variation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pierpont_variation&oldid=13447
This article was adapted from an original article by B.I. Golubov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article