Darboux property for transformations

Abstract

Numerous authors have given various notions of Darboux property for transformations defined on spaces more general than the real line. Among them, Neugebauer has defined a Darboux property for real-valued functions in a Euclidean space in terms of his Darboux sets and proved that derivatives of certain interval functions possess the Darboux property [S 1; Bruckner and Bruckner have introduced a notion of Darboux property for transformations from a Euclidean space to a metric space relative to some preassigned base for the domain and showed that certain known theorems on Darboux functions are also valid in their general setting [ 1 ]. Being motivated by the work in [ 11, we study in this article another notion of Darboux transformations. A local characterization of Darboux transformations, which generalizes a result found in [2], and a necessary and sufficient condition that a Baire type 1 transformation possesses the Darboux property are obtained. Moreover, it is shown that the above result concerning derivatives of interval functions and a theorem on approximately continuous transformations by Goffman and Waterman [3] may follow as our special cases. Throughout this paper, we shall use X to denote a Euclidean space, p the usual metric on X and X* a metric space with metric p*. If A (A *) is a subset of X (X*), then x (x*) is the closure of A (A *) in X (X*). Also, ..d is a (topological) base of connected sets for the space X such that any translation of any set in 9 is also in 59. It is found in [l] that a transformation f on X to X* is defined to be Darboux (9) if f(o) is connected for every U E 9. Now we give another notion: