Baire one star function
A Baire one star function is a type of function studied in real analysis. A function [math]\displaystyle{ f: \mathbb{R} \to \mathbb{R} }[/math] is in class Baire* one, written [math]\displaystyle{ f \in \mathbf{B}^{*}_{1} }[/math], and is called a Baire one star function, if for each perfect set [math]\displaystyle{ P \in \mathbb{R} }[/math], there is an open interval [math]\displaystyle{ I \in \mathbb{R} }[/math], such that [math]\displaystyle{ P \cap I }[/math] is nonempty, and the restriction [math]\displaystyle{ f |_{P \cap I} }[/math] is continuous. The notion seems to have originated with B. Kirchheim 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).
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