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On sets where lip f is finite
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Buczolich, Zoltan, Hanson, Bruce, Rmoutil, Martin and Zurcher, Thomas (2019) On sets where lip f is finite. Studia Mathematica, 249 (1). pp. 33-58. doi:10.4064/sm170820-26-5 ISSN 0039-3223.
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WRAP-On-sets-where-lip-f-is-finite-Zurcher-2018.pdf - Accepted Version Embargoed item. Restricted access to Repository staff only - Requires a PDF viewer. Download (752Kb) |
Official URL: https://doi.org/10.4064/sm170820-26-5
Abstract
Given a function f : R -> R, the so-called "little lip" function lip f is defined as follows: lip f(x) = lim inf(r SE arrow 0) sup(vertical bar x - y vertical bar <= r) vertical bar f(y) - f(x)vertical bar/r. We show that if f is continuous on R, then the set where lip f is infinite is a countable union of countable intersections of closed sets (that is, an F-sigma delta set). On the other hand, given a countable union E of closed sets, we construct a continuous function f such that lip f is infinite exactly on E. A further result is that, for a typical continuous function f on the real line, lip f vanishes almost everywhere.
Item Type: | Journal Article | ||||||
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Subjects: | Q Science > QA Mathematics | ||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||
Journal or Publication Title: | Studia Mathematica | ||||||
Publisher: | Polska Akademia Nauk | ||||||
ISSN: | 0039-3223 | ||||||
Official Date: | 26 April 2019 | ||||||
Dates: |
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Volume: | 249 | ||||||
Number: | 1 | ||||||
Page Range: | pp. 33-58 | ||||||
DOI: | 10.4064/sm170820-26-5 | ||||||
Status: | Peer Reviewed | ||||||
Publication Status: | Published | ||||||
Access rights to Published version: | Open Access (Creative Commons) | ||||||
Date of first compliant deposit: | 10 May 2018 | ||||||
Related URLs: | |||||||
Open Access Version: |
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