Sets of discontinuities for functions continuous on flats (Q2510988)

Some new information on some classical problems in analysis regarding the definition of continuity for functions of several variables, is given. Consider a family \(F\) of flats (affine subspaces of \(\mathbb R^n\)). In this paper the classes of \(F\)-continuous functions \(f:\mathbb R^n \to \mathbb R\) that are continuous when restricted to each subset in \(F\) are studied. In these terms, several definitions of partial continuity can be described. For example, if of all \(k \leq n\) dimensional flats are considered, we get the classical linear continuity, and taking affine subspaces of one dimension that are parallel to the axes, we get the separately continuous functions. The authors give a complete description of the points of discontinuity \(D(F)\) of \(F\)-continuous functions that is given if we take \(F\) as the set of parallel subspaces to the ones generated by \(k\) coordinate vectors, denoted by \(F_k\), providing also some properties of these sets (Theorem 2.1). They give some structural results on the collections of points of discontinuity \(D(F_k)\) (Theorem 2.2) and a characterization for \(k \geq n/2\) in terms of countable unions of some special compact sets (Theorem 2.3). An application to a 60 years old problem of Kronrod for the class \(D(F_1)\) is provided.