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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Darboux Baire-$.5$ functions
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by Harvey Rosen
Proc. Amer. Math. Soc. 110 (1990), 285-286
DOI: https://doi.org/10.1090/S0002-9939-1990-1017851-9

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.
References
  • Andrew M. Bruckner, Differentiation of real functions, Lecture Notes in Mathematics, vol. 659, Springer, Berlin, 1978. MR 507448
  • F. Burton Jones and E. S. Thomas Jr., Connected $G_{\delta }$ graphs, Duke Math. J. 33 (1966), 341–345. MR 192477
  • K. Kuratowski, Topology. Vol. I, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1966. New edition, revised and augmented; Translated from the French by J. Jaworowski. MR 0217751
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Bibliographic Information
  • © Copyright 1990 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 110 (1990), 285-286
  • MSC: Primary 26A21; Secondary 26A15, 54C08
  • DOI: https://doi.org/10.1090/S0002-9939-1990-1017851-9
  • MathSciNet review: 1017851