The Lipschitz Structure of Continuous Self-Maps of Generic Compact Sets
Continuous self-maps of closed sets generic with respect to the Hausdorff metric admit only a trivial Lipschitz structure. Unless ƒ is the identity on some nonempty open set of E, the image of any set on which ƒ is Lipschitz is nowhere dense in E. The set of points of differentiability of ƒ in E map...
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Veröffentlicht in: | Journal of mathematical analysis and applications 1994-12, Vol.188 (3), p.798-808 |
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Format: | Artikel |
Sprache: | eng |
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Zusammenfassung: | Continuous self-maps of closed sets generic with respect to the Hausdorff metric admit only a trivial Lipschitz structure. Unless ƒ is the identity on some nonempty open set of E, the image of any set on which ƒ is Lipschitz is nowhere dense in E. The set of points of differentiability of ƒ in E maps onto a first category subset of E. We apply these results and related ones to the study of omega-limit sets of continuous functions. We show that while all nonvoid nowhere dense closed sets are ω-limit sets for continuous functions, most closed sets are not ω-limit sets for functions, most closed sets are not ω-limit sets for functions exhibiting even minimal smoothness. |
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ISSN: | 0022-247X 1096-0813 |
DOI: | 10.1006/jmaa.1994.1463 |