Baire one star function is a term from real analysis. A function f : R → R is in class Baire* one, written f ∈ B 1 ∗ , and is called a Baire one star function, if for each perfect set P ∈ R , there is an open interval I ∈ R , such that P ∩ I is nonempty, and the restriction f | P ∩ 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).
