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 ?

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).

On the Theory of Fractional Calculus in the Pettis-Function Spaces

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

Multiresolution Analysis Applied to the Monge-Kantorovich Problem

an absolutely continuous function which satisfies (5.1) almost everywhere.

IMPLICIT DIFFERENTIAL INCLUSIONS WITH ACYCLIC RIGHT-HAND SIDES: AN ESSENTIAL FIXED POINTS APPROACH

The following identity for an absolutely continuous function f: [a, b] [right arrow] R holds (see [14]):

Some Hermite-Hadamard-Fejer Type Integral Inequalities for Differentiable [eta]-Convex Functions with Applications

It can be seen w'(t) is absolutely continuous function in [0,1].

Convergence Order of the Reproducing Kernel Method for Solving Boundary Value Problems

Then there exists an absolutely continuous function p and [mu] [member of] [C.sup.*] (0,1) such that

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

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

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).

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

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).

Interconnections between multiplier methods and window methods in generalized sampling

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.

Overview of viability results/olevaade vitaalsustulemustest

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

We will denote by [H.sub.w] the set of all absolutely continuous functions f defined on [0, [infinity]) satisfying f (0) = 0 and

Numerical Real Inversion of the Laplace Transform by Using Multiple-Precision Arithmetic