Jump function

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One of the three components in the Lebesgue decomposition of a function of bounded variation. Let be a function of bounded variation on an interval . Let when and when . Then is called the jump of at from the left and the jump of at from the right. If , then

is called the jump of at . Let be the sequence of all points of discontinuity of on and put

Then is called the jump function of . Note that the difference is a continuous function of bounded variation on and that , where is the variation of on (cf. Variation of a function). Moreover,


[1] H. Lebesgue, "Leçons sur l'intégration et la récherche des fonctions primitives" , Gauthier-Villars (1928)
[2] I.P. Natanson, "Theorie der Funktionen einer reellen Veränderlichen" , H. Deutsch , Frankfurt a.M. (1961) (Translated from Russian)


The function is also called the saltus function of . A function of bounded variation that equals its jump function is itself often called a jump function.


[a1] B. Szökefalvi-Nagy, "Introduction to real functions and orthogonal expansions" , Oxford Univ. Press (1965)
[a2] S. Saks, "Theory of the integral" , Hafner (1952) (Translated from French)
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Jump 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