Baire one star function

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 $f{|}_{P\cap I}$ 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).

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