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## Fundamenta Mathematicae

2001 | 168 | 2 | 119-130
Tytuł artykułu

### On nowhere weakly symmetric functions and functions with two-element range

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EN
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EN
A function f: ℝ → {0,1} is weakly symmetric (resp. weakly symmetrically continuous) at x ∈ ℝ provided there is a sequence hₙ → 0 such that f(x+hₙ) = f(x-hₙ) = f(x) (resp. f(x+hₙ) = f(x-hₙ)) for every n. We characterize the sets S(f) of all points at which f fails to be weakly symmetrically continuous and show that f must be weakly symmetric at some x ∈ ℝ∖S(f). In particular, there is no f: ℝ → {0,1} which is nowhere weakly symmetric.
It is also shown that if at each point x we ignore some countable set from which we can choose the sequence hₙ, then there exists a function f: ℝ → {0,1} which is nowhere weakly symmetric in this weaker sense if and only if the continuum hypothesis holds.
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Tom
Numer
Strony
119-130
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wydano
2001
Twórcy
autor
• Department of Mathematics, West Virginia University, Morgantown, WV 26506-6310, U.S.A.
autor
• Department of Mathematics, University of Wisconsin-Oshkosh, Oshkosh, WI 54901-8601, U.S.A.
autor
• Institute of Mathematics, Polish Academy of Sciences, Abrahama 18, Sopot, Poland
• Department of Mathematics, Gdańsk University, Wita Stwosza 57, 80-952 Gdańsk, Poland
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