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

Krzysztof Ciesielski, Kandasamy Muthuvel, and Andrzej Nowik. "On nowhere weakly symmetric functions and functions with two-element range." Fundamenta Mathematicae 168.2 (2001): 119-130. <http://eudml.org/doc/282063>.

@article{KrzysztofCiesielski2001,
abstract = { 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. },
author = {Krzysztof Ciesielski, Kandasamy Muthuvel, Andrzej Nowik},
journal = {Fundamenta Mathematicae},
keywords = {continuum hypothesis; second category sets; two-element range functions; symmetric functions; symmetrically continuous functions; anti-Schwartz functions},
language = {eng},
number = {2},
pages = {119-130},
title = {On nowhere weakly symmetric functions and functions with two-element range},
url = {http://eudml.org/doc/282063},
volume = {168},
year = {2001},
}

TY - JOUR
AU - Krzysztof Ciesielski
AU - Kandasamy Muthuvel
AU - Andrzej Nowik
TI - On nowhere weakly symmetric functions and functions with two-element range
JO - Fundamenta Mathematicae
PY - 2001
VL - 168
IS - 2
SP - 119
EP - 130
AB - 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.
LA - eng
KW - continuum hypothesis; second category sets; two-element range functions; symmetric functions; symmetrically continuous functions; anti-Schwartz functions
UR - http://eudml.org/doc/282063
ER -