On \(\mathbb{I}\)-approximate limits and \(\mathbb{I}\)-approximate smoothness

Rafał Zduńczyk

Commentationes Mathematicae, Tome 46 (2006) no. 1, pp. 1-16.

Résumé

In this paper we present some results based on slightly modified idea of the \(\mathbb{I}\)-density introduced by Władysław Wilczyński. Some theorems are generalized versions of results from [2] and [3]. We investigate properties of functions from \(\mathbb{R}^X\), where \(X\) is supplied with the \(\mathbb{I}\)-density. We try to free our considerations from the assumption of Baire property, or measurability. In some cases this is not done yet. Star-marked statements still need that assumption, proofs presented here are done for Baire property, but it is possible to adapt them to measure. \(\mathbb{I}\)-density itself does not require any structure of considered space but a metric vector space over \(\mathbb{R}\). However, in last section we confine ourselves to \(\mathbb{R}\), for we make use of \(\mathbb{R}\)’s structure for simplicity. To find more about related topics see [4], [5], more bibliography one can find in [1] and [5].

Détail

DOI : 10.14708/cm.v46i1.5204

Mots-clés : Density point, algebra of sets, generalized derivative