Baire one star function

A Baire one star function is a type of function studied in 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. 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).