Negative variation of a function

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negative increment of a function

One of the two terms whose sum is the complete increment or variation of a function on a given interval. Let be a function of a real variable, defined on an interval and taking finite real values.

Let be an arbitrary partition of and let

where the summation is over those numbers for which the difference is non-positive. The quantity

is called the negative variation (negative increment) of the function on the interval . It is always true that . See also Positive variation of a function; Variation of a function.


[1] H. Lebesgue, "Leçons sur l'intégration et la récherche des fonctions primitives" , Gauthier-Villars (1928)
How to Cite This Entry:
Negative 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