Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Darboux Baire-$ .5$ functions

Author: Harvey Rosen
Journal: Proc. Amer. Math. Soc. 110 (1990), 285-286
MSC: Primary 26A21; Secondary 26A15, 54C08
MathSciNet review: 1017851
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Abstract: Let $ I = [0,1]$, and let $ D$ denote the points of continuity of a function $ f:I \to R$. A Darboux function maps each connected set to a connected set. A function is Baire-$ 1$ (Baire-$ .5$) if preimages of open sets are $ {F_\sigma }$-sets ( $ {G_\delta }$-sets). We show that if $ f$ is a Darboux Baire-$ .5$ function, then the graph of the restriction of $ f$ to $ D$ is a dense subset of the whole graph of $ f$. It is already known that there is a Darboux Baire-$ 1$ function which does not satisfy this conclusion.

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Keywords: Darboux Baire-$ .5$ function, set of points of continuity
Article copyright: © Copyright 1990 American Mathematical Society