Kempisty's theorem for the integral product quasicontinuity

Zbigniew Grande¹

¹ Institute of Mathematics Kazimierz Wielki University Plac Weyssenhoffa 11 85-072 Bydgoszcz, Poland

Colloquium Mathematicum, Tome 106 (2006) no. 2, pp. 257-264.

Résumé

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.

Détail

DOI : 10.4064/cm106-2-6

Keywords: function mathbb mathbb satisfies condition resp point each real each set euclidean topology nbsp mathbb resp strong density topology ordinary density topology there set cap neq emptyset cap int cap kempistys theorem concerning product quasicontinuity investigated above notions