On continuity of symmetric restrictions of Borel functions
Author:
Michał Morayne
Journal:
Proc. Amer. Math. Soc. 93 (1985), 440442
MSC:
Primary 26B05; Secondary 26A24
MathSciNet review:
773998
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Abstract: We prove that if is a complete metric space denseinitself, is a compact metric space and is a Borelmeasurable 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 .
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John
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(30 #3855)
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 A. M. Bruckner, J. G. Ceder and M. L. Weiss, On differentiability structure of real functions, Trans. Amer. Math. Soc. 142 (1969), 113. MR 0259037 (41:3679)
 [2]
 J. P. Burgess, A selector principle for equivalence relations, Michigan Math. J. 24 (1977), 6576. 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.
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 J. Mycielski, Independent sets in topological algebras, Fund. Math. 55 (1964), 139147 MR 0173645 (30:3855)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198507739981
PII:
S 00029939(1985)07739981
Article copyright:
© Copyright 1985
American Mathematical Society
