• Source: Baire one star function

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

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