# 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.
References in periodicals archive ?
If the pair (x, u) is a minimizer for the OCP, then there exists an absolutely continuous function p and [mu] [member of] [C.
out]]) and that q is absolutely continuous function excepted at [t.
It is easy to see that for any locally absolutely continuous function f: [a, b] [right arrow] R, we have the identity
respectively, and to seek sharp upper bounds for these distances in terms of different measure that can be associated with f, where f is restricted to particular classes of functions including functions of bounded variation, Lipschitzian, convex and absolutely continuous functions.
2) is an absolutely continuous function, which satisfies that equation on (0, [infinity]) almost everywhere and condition (1.
4) is defined as a locally absolutely continuous function x(t), with [?
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.
for x [member of] [a, b] where f : [a, b] [right arrow] R is an absolutely continuous function on [a, b].
2]) [member of] (0,1] x (0,1], f : J x R [right arrow] R, is a given function and [phi] : [0,a] [right arrow] R, [psi] : [0,b] [right arrow] R are given absolutely continuous functions with [phi](0) = [psi](0).
The rate is best possible amongst absolutely continuous functions f on [-1, 1] whose derivative is bounded.
We denote the class of absolutely continuous functions in Caratheodory's sense by [C.
loc] refers to the class of locally absolutely continuous functions.
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