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
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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.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
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