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
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is called the jump of at
. Let
be the sequence of all points of discontinuity of
on
and put
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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,
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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) |
Jump function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Jump_function&oldid=27826