Sums and products of quasi-continuous functions

Abstract

In this article two main results are proved. The first one is that each cliquish function \(f\colon \mathbb{R}^k \to \mathbb{R} \) is the sum of two quasi-continuous functions. It is also shown that we can moreover require that the summands preserve points of continuity of \(f\), are bounded provided that \(f\) is bounded and belong to the same class of Baire as \(f\) (if \(f\) is Borel measurable). The other main result is that each function \(f\colon \mathbb{R}^k \to \mathbb{R} \) which can be written as the product of finitely many quasi-continuous functions, can be expressed as the product of two quasi-continuous functions, and we can require that the factors belong to the same class of Baire as \(f\) (if \(f\) is Borel measurable).