Kempisty's theorem for the integral product quasicontinuity

Volume 106 / 2006

Zbigniew Grande Colloquium Mathematicum 106 (2006), 257-264 MSC: 26B05, 26A03, 26A15. DOI: 10.4064/cm106-2-6

Abstract

A function $f:\mathbb R ^n \to \mathbb R$ satisfies the condition $Q_i(x)$ (resp. $Q_s(x)$, $Q_o(x)$) at a point $x$ if for each real $r > 0$ and for each set $U \ni x$ open in the Euclidean topology of $\mathbb R^n$ (resp. strong density topology, ordinary density topology) there is an open set $I$ such that $I \cap U \neq \emptyset $ and $|(1/\mu (U\cap I))\int_{U \cap I} f(t)\,dt - f(x)| < r$. Kempisty's theorem concerning the product quasicontinuity is investigated for the above notions.

Authors

  • Zbigniew GrandeInstitute of Mathematics
    Kazimierz Wielki University
    Plac Weyssenhoffa 11
    85-072 Bydgoszcz, Poland
    e-mail

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