On Relations Among Various Classes of I-a.e. Continuous Darboux Functions

Abstract

This paper is devoted to relationships among various classes of $\cal I$-a.e. continuous functions (i.e., of functions whose sets of discontinuity points belong to certain $\sigma$-ideals $\cal I$ consisting of boundary sets). For instance, if $\cal K$ is the $\sigma$-ideal of first category sets and $\cal I$ denotes the $\sigma$-ideal of all sets that are: of Lebesque measure zero, $\sigma$-porous, or countable, then the set of $\cal I$-a.e. continuous functions is uniformly porous in the space of all $\cal K$-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.