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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Typical continuous functions are virtually nonmonotone


Authors: P. Humke and M. Laczkovich
Journal: Proc. Amer. Math. Soc. 94 (1985), 244-248
MSC: Primary 26A15; Secondary 26A48
MathSciNet review: 784172
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Abstract: For every porosity premeasure $ \phi $, a typical continuous function meets every monotone function in a bilaterally strongly $ \phi $-porous set. The statement does not remain valid if we replace the class of monotone functions by the class of absolutely continuous functions.


References [Enhancements On Off] (What's this?)

  • [H] J. Haussermann, Generalized porosity characteristics of a residual set of continuous functions, Ph. D. dissertation, University of California, Santa Barbara, 1984.
  • [T] B. S. Thomson, On the level set structure of a continuous function, Classical real analysis (Madison, Wis., 1982) Contemp. Math., vol. 42, Amer. Math. Soc., Providence, RI, 1985, pp. 187–190. MR 807990 (87h:26002), http://dx.doi.org/10.1090/conm/042/807990

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1985-0784172-7
PII: S 0002-9939(1985)0784172-7
Keywords: Typical continuous functions, intersections with monotone functions, porous sets
Article copyright: © Copyright 1985 American Mathematical Society