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

Fundamenta Mathematicae (2001)

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

top

Abstract

top
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.

How to cite

top

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 -

NotesEmbed?

top

You must be logged in to post comments.

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

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.