Baire one star function explained

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

f: R \to R is in class Baire* one, written f \in B ^{*} _{1} , and is called a Baire one star function if, for each perfect set P \in R , there is an open interval I \in R , such that P \cap I is nonempty, and the restriction f | _{P} 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).