Oscillation of a function
From Encyclopedia of Mathematics
on a set
The difference between the least upper and the greatest lower bounds of the values of on
. In other words, the oscillation of
on
is given by
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If the function is unbounded on , its oscillation on
is put equal to
. For constant functions on
(and only for these) the oscillation on
is zero. If the function
is defined on a subset
of
, then its oscillation at any point
of the closure of
is defined by the formula
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where the infimum is taken over all neighbourhoods of
. If
, then in order that
be continuous at
with respect to the set
it is necessary and sufficient that
.
Comments
The function is called the oscillation function of
.
References
[a1] | K.R. Stromberg, "Introduction to classical real analysis" , Wadsworth (1981) pp. 120 |
[a2] | R.R. Goldberg, "Methods of real analysis" , Blaisdell (1964) pp. 129 |
How to Cite This Entry:
Oscillation of a function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Oscillation_of_a_function&oldid=48086
Oscillation of a function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Oscillation_of_a_function&oldid=48086
This article was adapted from an original article by A.A. Konyushkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article