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

✍ Scribed by Krzysztof Ciesielski; Jan Jastrzȩbski

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 168 KB
• Volume: 103
• Category: Article
• ISSN: 0166-8641
• DOI: 10.1016/s0166-8641(98)00169-2

✦ Synopsis

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 R n 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 R 2 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).)