On ℰ-continuous functions

Abstract

Some properties of \({\mathcal E}\)-continuous functions are investigated. In particular, the maximal family with respect to outer and inner compositions for the family of all \({\mathcal E}\)-continuous functions are described. Moreover, under some assumptions on \({\mathcal E}\) it is proved that every function \(f:\mathbb{R} \rightarrow \mathbb{R} \) can be represented as the composition of two \({\mathcal E}\)-continuous function. Similarly, every function \(f:\mathbb{R} \rightarrow \mathbb{R} \) can be represented as the limit of a transfinite sequence of \({\mathcal E}\)-continuous functions.