Real Analysis Exchange

On Riemann integral quasicontinuity.

Zbigniew Grande

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Abstract

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

Article information

Source
Real Anal. Exchange, Volume 31, Number 1 (2005), 239-252.

Dates
First available in Project Euclid: 5 June 2006

Permanent link to this document
https://projecteuclid.org/euclid.rae/1149516802

Mathematical Reviews number (MathSciNet)
MR2218200

Zentralblatt MATH identifier
1107.26005

Subjects
Primary: 26A05 26A15: Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) {For properties determined by Fourier coefficients, see 42A16; for those determined by approximation properties, see 41A25, 41A27}

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

Citation

Grande, Zbigniew. On Riemann integral quasicontinuity. Real Anal. Exchange 31 (2005), no. 1, 239--252. https://projecteuclid.org/euclid.rae/1149516802


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