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Proceedings of the American Mathematical Society

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On continuity of symmetric restrictions of Borel functions

Author: Michał Morayne
Journal: Proc. Amer. Math. Soc. 93 (1985), 440-442
MSC: Primary 26B05; Secondary 26A24
MathSciNet review: 773998
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Abstract: We prove that if $ X$ is a complete metric space dense-in-itself, $ Y$ is a compact metric space and $ F:X \times X\backslash \left\{ {(x,x):x \in X} \right\} \to Y$ is a Borel-measurable function such that $ F({x_1},{x_2}) = F({x_2},{x_1})$ for every $ {x_1},{x_2} \in X,{x_1} \ne {x_2}$, then there is a perfect subset $ P$ of $ X$ such that $ F$ is uniformly continuous on $ P \times P\backslash \left\{ {(x,x):x \in P} \right\}$. An immediate corollary of the above result is the following theorem proved by Bruckner, Ceder and Weiss: If $ F$ is a real continuous function defined on a perfect set $ P \subset R$, there is a perfect subset $ Q$ of $ P$ such that $ f\vert Q$ has a derivative (finite or infinite) at every point of $ Q$.

References [Enhancements On Off] (What's this?)

  • [1] A. M. Bruckner, J. G. Ceder and M. L. Weiss, On differentiability structure of real functions, Trans. Amer. Math. Soc. 142 (1969), 1-13. MR 0259037 (41:3679)
  • [2] J. P. Burgess, A selector principle for $ \Sigma _1^1$ equivalence relations, Michigan Math. J. 24 (1977), 65-76. MR 0453530 (56:11792)
  • [3] F. Galvin, Partition theorems for the real line, Notices Amer. Math. Soc. 15 (1968), 660.
  • [4] -, Errata to "Partition theorems for the real line", Notices Amer. Math. Soc. 16 (1969), 1095.
  • [5] J. Mycielski, Independent sets in topological algebras, Fund. Math. 55 (1964), 139-147 MR 0173645 (30:3855)

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