A Baire one star function is a type of function studied in real analysis. A function f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } is in class Baire* one, written f ∈ B 1 ∗ {\displaystyle f\in \mathbf {B} _{1}^{*}} , and is called a Baire one star function if, for each perfect set P ∈ R {\displaystyle P\in \mathbb {R} } , there is an open interval I ∈ R {\displaystyle I\in \mathbb {R} } , such that P ∩ I {\displaystyle P\cap I} is nonempty, and the restriction f | P ∩ I {\displaystyle 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).
References Maliszewski, Aleksander (1998), "On the averages of Darboux functions", Transactions of the American Mathematical Society, 350 (7): 2833–2846, doi:10.1090/S0002-9947-98-02267-3