Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

Sierpiński-Zygmund functions that are Darboux, almost continuous, or have a perfect road

Sierpiński-Zygmund functions that are Darboux, almost continuous, or have a perfect road In this paper we show that if the real line ${\Bbb R}$ is not a union of less than continuum many of its meager subsets then there exists an almost continuous Sierpiński–Zygmund function having a perfect road at each point. We also prove that it is consistent with ZFC that every Darboux function $f\colon{\Bbb R}\to{\Bbb R}$ is continuous on some set of cardinality continuum. In particular, both these results imply that the existence of a Sierpiński–Zygmund function which is either Darboux or almost continuous is independent of ZFC axioms. This gives a complete solution of a problem of Darji [4]. The paper contains also a construction (in ZFC) of an additive Sierpiński–Zygmund function with a perfect road at each point. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Archive for Mathematical Logic Springer Journals

Sierpiński-Zygmund functions that are Darboux, almost continuous, or have a perfect road

Download PDF
7 pages

Loading next page...
 
/lp/springer-journals/sierpi-ski-zygmund-functions-that-are-darboux-almost-continuous-or-cze0hVp2jO

References (24)

Publisher
Springer Journals
Copyright
Copyright © 1997 by Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Mathematical Logic and Foundations; Mathematics, general; Algebra
ISSN
0933-5846
eISSN
1432-0665
DOI
10.1007/s001530050080
Publisher site
See Article on Publisher Site

Abstract

In this paper we show that if the real line ${\Bbb R}$ is not a union of less than continuum many of its meager subsets then there exists an almost continuous Sierpiński–Zygmund function having a perfect road at each point. We also prove that it is consistent with ZFC that every Darboux function $f\colon{\Bbb R}\to{\Bbb R}$ is continuous on some set of cardinality continuum. In particular, both these results imply that the existence of a Sierpiński–Zygmund function which is either Darboux or almost continuous is independent of ZFC axioms. This gives a complete solution of a problem of Darji [4]. The paper contains also a construction (in ZFC) of an additive Sierpiński–Zygmund function with a perfect road at each point.

Journal

Archive for Mathematical Logic – Springer Journals

Published: Dec 1, 1997

There are no references for this article.