Baire one star function

Baire one star function is a term from real analysis. A function $f: \mathbb{R} \to \mathbb{R}$ is in class Baire* one, written $f \in \mathbf{B}^{*}_{1}$, and is called a Baire one star function, if for each perfect set $P \in \mathbb{R}$, there is an open interval $I \in \mathbb{R}$, such that $P \cap I$ is nonempty, and the restriction $f |_{P \cap I}$ is continuous. The notion seems to have originated with B. Kerchheim in an article titled 'Baire one star functions' (Real Anal. Exch. 18 (1992/93), 385-399).

The terminology is actually due to Richard O'Malley,  'Baire* 1, Darboux functions' Proc. Amer. Math. Soc. 60 (1976) 187-192.  The concept itself (under a different name) goes back at least to 1951. See H. W. Ellis, 'Darboux properties and applications to nonabsolutely convergent integrals' Canad. Math. J., 3 (1951), 471-484, where the same concept is labelled as [CG] (for generalized continuity).

External links

• A paper which defines class Baire* one