FIRST RETURN APPROACHABILITY

We consider real-valued functions defined on [0, 1]. Both the class of Baire one functions and the class of Baire-one, Darboux functions hav e been characterized using first return limit notions. The former clas s consists of the first return recoverable functions and the latter co nsists of the first return continuous functions. Here we introduce a n atural intermediate type of first return limiting process, first retur n approachability, and show that the first return approachable functio ns are precisely those Baire class one functions whose graphs are dens e in themselves. Also, the set of points at which a function is first return approachable, but not first return continuous, is shown to be s igma-porous. (C) 1996 Academic Press, Inc.