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2005-2006 On Riemann integral quasicontinuity.
Zbigniew Grande
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Real Anal. Exchange 31(1): 239-252 (2005-2006).


A function $f : \mathbb{R}^n \to \mathbb{R}$ satisfies condition $(Q_{r,i}(x))$ (resp. ($Q_{r,s}(x))$, [$Q_{r,o}(x)$]) at a point $x$ if for each real $r > 0$ and for each set $U$ containing $x$ and belonging to Euclidean topology in $\mathbb{R} ^n$ (resp. to the strong density topology [to the ordinary density topology]) there is a regular domain $I$ such that int$(I) \cap U \neq \emptyset $, $f\!\restriction\!I$ is integrable in the sense of Riemann and $|\frac{1}{\mu (U\cap I)}\int_{U \cap I} f(t)\,dt - f(x)| < r$. These notions are particular cases of their analogues for the Lebesgue integral. In this article we compare these notions with the classical quasicontinuity and integral quasicontinuities


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Zbigniew Grande. "On Riemann integral quasicontinuity.." Real Anal. Exchange 31 (1) 239 - 252, 2005-2006.


Published: 2005-2006
First available in Project Euclid: 5 June 2006

zbMATH: 1107.26005
MathSciNet: MR2218200

Primary: 26A05 , 26A15

Keywords: density topology , functions of two variables , Riemann integral quasicontinuity , uniform limit

Rights: Copyright © 2005 Michigan State University Press

Vol.31 • No. 1 • 2005-2006
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