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

Krzysztof Ciesielski; Kandasamy Muthuvel; Andrzej Nowik

Fundamenta Mathematicae (2001)

- Volume: 168, Issue: 2, page 119-130
- ISSN: 0016-2736

## Access Full Article

top## Abstract

top## How to cite

topKrzysztof 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 -

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.