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Positive variation of a function

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One of the two terms whose sum is the complete change, or (total) variation, of the function (cf. Variation of a function) over a given interval. Let be a function of a real variable given on the segment and taking real values. Let be any partition of and let

where the summation is over those values of for which the difference is non-negative. The quantity

is called the positive variation of the function over . Of course, . The concept of the positive variation of a function was introduced by C. Jordan [1]. See also Negative variation of a function.

References

[1] C. Jordan, "Sur la série de Fourier" C.R. Acad. Sci. Paris , 92 (1881) pp. 228–230
[2] H. Lebesgue, "Leçons sur l'intégration et la récherche des fonctions primitives" , Gauthier-Villars (1928)
How to Cite This Entry:
Positive variation of a function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Positive_variation_of_a_function&oldid=27950
This article was adapted from an original article by B.I. Golubov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article