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

• Autorzy: Krzysztof Ciesielski , Kandasamy Muthuvel , Andrzej Nowik
• Języki publikacji: EN

Abstrakty

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.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2001
• Tom: 168
• Numer: 2
• Strony: 119-130

Daty

wydano

2001

Twórcy

autor

Krzysztof Ciesielski

• Department of Mathematics, West Virginia University, Morgantown, WV 26506-6310, U.S.A.

autor

Kandasamy Muthuvel

• Department of Mathematics, University of Wisconsin-Oshkosh, Oshkosh, WI 54901-8601, U.S.A.

autor

Andrzej Nowik

• Institute of Mathematics, Polish Academy of Sciences, Abrahama 18, Sopot, Poland
• Department of Mathematics, Gdańsk University, Wita Stwosza 57, 80-952 Gdańsk, Poland

Identyfikatory

DOI

10.4064/fm168-2-3