Some remarks on \({\mathcal J}\)-approximately continuous functions (Q1071118)

The note deals with some problems concerning the class of \({\mathcal J}\)- approximately continuous functions, in the framework (R,S,\({\mathcal J})\), where, R is the real line, S is the \(\sigma\)-algebra of subsets of R having the Baire property, \({\mathcal J}\) is the \(\sigma\)-algebra of subsets of R having the Baire property, \({\mathcal J}\) is the \(\sigma\)-ideal of sets of the first category on R. The notions of \({\mathcal J}\)-density point and \({\mathcal J}\)-dispersion point of a set \(A\in S\) are defined. If \(\Phi\) (A) means the set \(\{x\in R:x\quad is\quad an\quad {\mathcal J}-density\quad point\quad of\quad A\},\) then the class \({\mathcal T}_{{\mathcal J}}=\{A\in S:A\subseteq \Phi (A)\}\) is a topology on R. We remark that there is no connection between notions of \({\mathcal J}\)-density and density in the ordinary sense, with respect to Lebesgue measure (see Theorem 1). The classes of: \({\mathcal J}\)-approximately continuous functions, approximately continuous functions, derivative functions and bounded derivative functions are compared (Theorems 2 and 3). In the final part of the note, an example of an \({\mathcal J}\)-approximately continuous function, \(f:R\to R,\) is constructed. Let E be the set of discontinuity points of the mentioned function f; then every point \(x_ 0\in E\) is an \({\mathcal J}\)-density point of the set \(\{x:f(x)=0\},\) and every point of E is an \({\mathcal J}\)-dispersion point of a certain set F (see Theorem 5).