On nowhere weakly symmetric functions and functions with two-element range
| Autor: | Krzysztof Ciesielski, Kandasamy Muthuvel, Andrzej Nowik |
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| Rok vydání: | 2001 |
| Předmět: | |
| Zdroj: | Fundamenta Mathematicae. 168:119-130 |
| ISSN: | 1730-6329 0016-2736 |
| DOI: | 10.4064/fm168-2-3 |
| Popis: | A function f : R →{ 0, 1} is weakly symmetric (resp. weakly symmetri- cally 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. 1. Introduction. The terminology of this note is standard and fol- lows (5). Recall that a function f : R → R is symmetrically continuous provided |
| Databáze: | OpenAIRE |
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