Proceedings of the American Mathematical Society

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



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$.

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Article copyright: © Copyright 1985 American Mathematical Society