Real Analysis Exchange

On Riemann integral quasicontinuity.

Zbigniew Grande

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

First available in Project Euclid: 5 June 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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}

Riemann integral quasicontinuity density topology uniform limit functions of two variables


Grande, Zbigniew. On Riemann integral quasicontinuity. Real Anal. Exchange 31 (2005), no. 1, 239--252.

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