Darboux-like functions within the classes of Baire one, Baire two, and additive functions
The authors consider relations between Darboux-like functions [see \textit{R.~G.~Gibson, T.~Natkaniec}, Real Anal. Exch. 22, No.~2, 492-533 (1996; Zbl 0942.26004)] within the classes of Baire one, Baire two, Borel and additive functions from \({\mathbb{R}}^n \to {\mathbb{R}}\). Main results: (1) There exists an almost continuous, Baire class two function \(f\colon {\mathbb{R}}\to {\mathbb{R}}\) which is not extendable. (2) There exists an additive extendable function \(f\colon {\mathbb{R}}\to {\mathbb{R}}\) which is not continuous. (3) For \(n>1\) every additive, almost continuous Darboux function \(f\colon \mathbb{R}^n\to \mathbb{R}\) is continuous.
scientific article; zbMATH DE number 1455643