Baire one star function

Baire one star function is a term from real analysis. A function $f:{\mathbb {R}}\to {\mathbb {R}}$ is in class Baire* one, written $f\in \mathbf {B} _{1}^{*}$ , and is called a Baire one star function, if for each perfect set $P\in \mathbb {R}$ , there is an open interval $I\in \mathbb {R}$ , such that $P\cap I$ is nonempty, and the restriction $f|_{P\cap I}$ is continuous. The notion seems to have originated with B. 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).

External links

A paper which defines class Baire* one

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.