Baire one star function

In mathematics, a function ${\displaystyle f:\mathbb {R} \to \mathbb {R} }$ is a Baire one star function, written ${\displaystyle f\in \mathbf {B} _{1}^{*}}$, if, for each perfect set ${\displaystyle P\in \mathbb {R} }$, there is an open interval ${\displaystyle I\in \mathbb {R} }$ such that ${\displaystyle P\cap I}$ is nonempty and the restriction ${\displaystyle f\,\mid _{P\cap I}}$ is continuous. The concept itself (under a different name) goes back at least to 1951, where it is labelled as [CG] (for generalized continuity).

References

Sources

• Ellis, H. W. (1951). "Darboux properties and applications to non-absolutely convergent integrals". Canadian Journal of Mathematics. 3: 471–485. doi:10.4153/CJM-1951-048-x.

• Kirchheim, Bernd (1993). "Baire one star functions". Real Analysis Exchange. 18 (2): 385–399. Zbl 853.54018.

• 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.

• O'Malley, Richard J. (1976). "Baire* 1, Darboux functions". Proceedings of the American Mathematical Society. 60 (1): 187–192. JSTOR 2041139.

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