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 is a complete metric space dense-in-itself, is a compact metric space and is a Borel-measurable function such that for every , then there is a perfect subset of such that is uniformly continuous on . An immediate corollary of the above result is the following theorem proved by Bruckner, Ceder and Weiss: If is a real continuous function defined on a perfect set , there is a perfect subset of such that has a derivative (finite or infinite) at every point of .

**[1]**A. M. Bruckner, J. G. Ceder, and Max L. Weiss,*On the differentiability structure of real functions*, Trans. Amer. Math. Soc.**142**(1969), 1–13. MR**0259037**, 10.1090/S0002-9947-1969-0259037-5**[2]**John P. Burgess,*A selector principle for ∑¹₁ equivalence relations*, Michigan Math. J.**24**(1977), no. 1, 65–76. MR**0453530****[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]**Jan Mycielski,*Independent sets in topological algebras*, Fund. Math.**55**(1964), 139–147. MR**0173645**

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DOI:
http://dx.doi.org/10.1090/S0002-9939-1985-0773998-1

Article copyright:
© Copyright 1985
American Mathematical Society