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).