Darboux-like functions within the classes of Baire one, Baire two, and additive functions

Abstract

In the paper we present an exhaustive discussion of the relations between Darboux-like functions within the classes of Baire one, Baire two, Borel, and additive functions from Rn into R. In particular we construct an additive extendable discontinuous function f:R→R, answering a question of Gibson and Natkaniec (1996–97, p. 499), and show that there is no similar function from R2 into R. We also describe a Baire class two almost continuous function f:R→R which is not extendable. This gives a negative answer to a problem of Brown, Humke, and Laczkovich (1988, Problem 1). (See also Problem 3.21 of Gibson and Natkaniec (1996–97).)