On relations among various classes of \({\mathcal I}\) a. e. continuous Darboux functions. (English) Zbl 1009.26012

Tomaszewska, Aneta

Real Anal. Exch. 25(1999-2000), No. 2, 695-702 (2000).

Summary: This paper is devoted to relationships among various classes of \({\mathcal I}\)-a.e. continuous functions (i.e., of functions whose sets of discontinuity points belong to certain \(\sigma\)-ideals \({\mathcal I}\) consisting of boundary sets). For instance, if \({\mathcal K}\) is the \(\sigma\)-ideal of first category sets and \({\mathcal I}\) denotes the \(\sigma\)-ideal of all sets that are: of Lebesgue measure zero, \(\sigma\)-porous, or countable, then the set of \({\mathcal I}\)-a.e. continuous functions is uniformly porous in the space of all \({\mathcal K}\)-a.e. continuous Darboux functions from \(\mathbb{R}^2\) into \(\mathbb{R}^2\) equipped with the metric of uniform convergence. As a tool in the proofs, symmetric Cantor sets in \(\mathbb{R}^2\) are used.

MSC:

26B05	Continuity and differentiation questions
26A15	Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
26A21	Classification of real functions; Baire classification of sets and functions

Keywords:

\(\sigma\)-ideals; uniformly porous sets; \({\mathcal I}\)-a.e. continuous functions; Darboux functions; symmetric Cantor sets