Universally First Return Continuous Functions

Abstract

Abstract.It is known that the first return continuous functions are precisely the Darboux functions in Baire class 1, and that every such function can be changed via a homeomorphism into an approximately continuous function. Here we give two characterizations of the smaller class of universally first return continuous functions, one of which is the capacity of changing such a function via a homeomorphism into an approximately continuous function which is continuous almost everywhere. O'Malley [4] introduced the notion of first return path systems to study differentiation properties of real-valued functions defined on the unit interval. In [1] the present authors showed that a large class of functions, including the approximately differentiable functions, are universally first return differentiable. In that same paper the notions of first return continuity and universal first return continuity were introduced, and first return continuous functions were shown to be the familiar Darboux, Baire 1 functions. Here we give two characterizations of the proper subclass of the Darboux, Baire 1 functions consisting of the universally first return continuous functions. We shall show that / : [0, 1] -► R is universally first return continuous if and only if / is Darboux, Baire 1 and the graph of / restricted to C(f), the set of points of continuity of /, is dense in the entire graph. We let & denote this collection of all Darboux, Baire 1 functions / : [0, 1] -> R for which f\C(f) is dense in /. Recalling that the Maximoff-Preiss Theorem (see [5], Theorem 2) states that a function is Darboux, Baire 1 if and only if there is a homeomorphism h of [0, 1 ] onto itself such that / o h is approximately continuous, it seems natural to inquire if we can obtain an even nicer function via a homeomorphism starting with a function in 3?, and perhaps characterizing the class S? in this manner. We show that this is, indeed, possible.Before proving our results, we need to review some terminology from [ 1 ] and [3]. By a trajectory we mean any sequence {x"}™=0 of distinct points in (0, 1), which is dense in [0,1]. One method of specifying a trajectory is to assign an enumeration or ordering to a given countable dense subset D of (0,1).