# Jump function

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,

#### References

[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) |

#### Comments

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.

#### References

[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) |

**How to Cite This Entry:**

Jump function.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Jump_function&oldid=18207