Darboux Baire-$.5$ functions

Abstract

Let I = [0, 1], and let D denote the points of continuity of a function f: I A. A Darboux function maps each connected set to a connected set. A function is BaireI (Baire.5 ) if prelmages of open sets are F.-sets (Gasets). We show that if f is a Darboux Baire-.5 function, then the graph of the restriction of f to D is a dense subset of the whole graph of f . It is already known that there is a Darboux Baire1 function which does not satisfy this conclusion. A classical result from real analysis states that the set D of points of continuity of an arbitrary function f: I -* R is a GI3-set. In 1966, Jones and Thomas showed that for any function f: I -I with a connected G3,-graph, D is also a dense subset of I [2]. Their argument still works for a Darboux function with a GQ-graph. It is not always the case that a Darboux Baire-1 or even a bounded approximately continuous function f satisfies the stronger property that each point on the graph of f has a point of f D plotted nearby [1, Chapter II, Theorems 1, 2.4, and 6.5]. However, we show this property is satisfied by the Darboux Baire-.5 functions, which form a subcollection of the Darboux Baire-1 functions. For a subset A of B in the plane, we say that A is bilaterally c-dense in B if in each open neighborhood of any point (x, y) c B lie c-many points of A to the left and to the right of (x, y) . Theorem. Suppose f: I -R is a Darboux Baire-.5 function, and let D denote the set of points at which f is continuous. Then the graph of f ID is bilaterally c-dense in the graph of f . Proof. Since the graph of f is a GI,-set, it is the intersection of a nested sequence of open subsets G1, G2. ... of I x R. By [2], D is a dense G3subset of I. We first show that the graph of f D is dense in the whole graph of f. Assurne it is not. It follows that there is an open neighborhood S (al, b1) x (cl, dl) of a point (x0, f(xo)) in I x R such that Received by the editors April 13, 1989 and, in revised form, July 28, 1989; presented at the Sixth Annual Miniconference on Real Analysis at Auburn University, April 8. 1989. 1980 Mathemnatics Subject Classification (1985 Revision). Prinmary 54C08; Secondary 26A15. Keyv words and phrases. Darboux Baire.5 function, set of points of continuity. g 1990 American Mathematical Society 0002-9939/90 $1.00 + $.25 per page