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Proceedings of the American Mathematical Society
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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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1990-1017851-9
PII: S 0002-9939(1990)1017851-9
Keywords: Darboux Baire-$ .5$ function, set of points of continuity
Article copyright: © Copyright 1990 American Mathematical Society