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), … , (a_n, b_n), 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.

Optimal Control Applied to an Irrigation Planning Problem

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.

A companion for the generalized Ostrowski and the generalized trapezoid type inequalities

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 [?

The [L.sup.p]-version of the generalized Bohl-Perron principle for vector equations with infinite delay

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.

Interconnections between multiplier methods and window methods in generalized sampling

for x [member of] [a, b] where f : [a, b] [right arrow] R is an absolutely continuous function on [a, b].

Ostrowski-Gruss-Cebysev type inequalities involving several functions

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).

Upper and lower solutions method for partial hyperbolic differential equations with Caputo's fractional derivative

The rate is best possible amongst absolutely continuous functions f on [-1, 1] whose derivative is bounded.

Bernstein's weighted approximation on R still has problems

We denote the class of absolutely continuous functions in Caratheodory's sense by [C.

An application of Fryszkowski's selection theorem to the Darboux Problem for third order hyperbolic inclusions

loc] refers to the class of locally absolutely continuous functions.

Generalized weighted Sobolev spaces and applications to Sobolev orthogonal polynomials: a survey *