MAKING BOREL FUNCTIONS LOOK CONTINUOUS

Abstract

We investigate the possibility of making all functions in Baire class , < ! 1, close to continuous by adding to them a single function of Baire class + 1. Remnants of are preserved, in various senses, in arbitrary Borel functions. Yet another way in which this is true has been considered recently by several authors. Given < ! 1, can we nd a single function f which makes all functions of the form f +g, with g Baire class , look continuous? Obviously, we would like f to be as simple as possible and the functions f +g as close to continuous as possible. The following four denitions are normally used, in this context, to measure the \degree of continuity of f +g. How to measure simplicity of f will be clear from the statements of results. Let g be a function mapping a separable metric space X into the reals. g is Darboux if images of connected sets are connected. g is connectivity if the graph of the restriction of g to any connected subset of X is a connected subset of X R. g is almost continuous if any open subset of X R containing the graph of g contains also the graph of a continuous function from X to R. g is extendable connectivity if g(x) = G(x; 0), for all x 2 X, for some connectivity function G : X [0; 1]! R. It is known that if X = R, then the above classes are getting properly smaller as we go down the list and that all of them contain the continuous functions.