# absolutely continuous function

## absolutely continuous function

[¦ab·sə‚lüt·lē kən¦tin·yə·wəs ′fəŋk·shən]
(mathematics)
A function defined on a closed interval with the property that for any positive number ε there is another positive number η such that, for any finite set of nonoverlapping intervals, (a1, b1), (a2, b2), … , (an, bn), whose lengths have a sum less than η, the sum over the intervals of the absolute values of the differences in the values of the function at the ends of the intervals is less than ε.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
References in periodicals archive ?
As shown in Example 23, there is an infinite dimension Banach space E and weakly absolutely continuous function y : I [right arrow] E which is nowhere weakly differentiable (hence the CFWD of y does not exist).
From the fact that X is compact, we obtain that f is absolutely continuous function on X; thus, given [member of] > 0, there exists [delta] > 0 such that
an absolutely continuous function which satisfies (5.1) almost everywhere.
The following identity for an absolutely continuous function f: [a, b] [right arrow] R holds (see [14]):
It can be seen w'(t) is absolutely continuous function in [0,1].
Then there exists an absolutely continuous function p and [mu] [member of] [C.sup.*] (0,1) such that
It is easy to see that for any locally absolutely continuous function f: [a, b] [right arrow] R, we have the identity
A solution of problem (1.1), (1.2) is an absolutely continuous function, which satisfies that equation on (0, [infinity]) almost everywhere and condition (1.2).
Lemma 2.2 Let p [member of] [1, [infinity]) and let g : R [right arrow] C be a locally absolutely continuous function with g and g' belonging to [L.sup.p](R).
Then for every ([t.sub.0]; [y.sub.0]) [member of] GrK the multivalued Cauchy problem (2.1) has a viable solution y : [[t.sub.0]; a] [right arrow] [R.sup.n] which is an absolutely continuous function.
for x [member of] [a, b] where f : [a, b] [right arrow] R is an absolutely continuous function on [a, b].
We will denote by [H.sub.w] the set of all absolutely continuous functions f defined on [0, [infinity]) satisfying f (0) = 0 and
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