Jump function
From Encyclopedia of Mathematics
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=27826
Jump function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Jump_function&oldid=27826
This article was adapted from an original article by B.I. Golubov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article