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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Nowhere monotone functions and functions of nonmonotonic type
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by Jack B. Brown, Udayan B. Darji and Eric P. Larsen PDF
Proc. Amer. Math. Soc. 127 (1999), 173-182 Request permission

Abstract:

We investigate the relationships between the notions of a continuous function being monotone on no interval, monotone at no point, of monotonic type on no interval, and of monotonic type at no point. In particular, we characterize the set of all points at which a function that has one of the weaker properties fails to have one of the stronger properties. A theorem of Garg about level sets of continuous, nowhere monotone functions is strengthened by placing control on the location in the domain where the level sets are large. It is shown that every continuous function that is of monotonic type on no interval has large intersection with every function in some second category set in each of the spaces $\mathcal {P}^n, C^n$, and $Lip^1$.
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Additional Information
  • Jack B. Brown
  • Affiliation: Department of Mathematics, Auburn University, Auburn, Alabama 36849-5310
  • Email: brownj4@mail.auburn.edu
  • Udayan B. Darji
  • Affiliation: Department of Mathematics, University of Louisville, Louisville, Kentucky 40292-0001
  • MR Author ID: 318780
  • ORCID: 0000-0002-2899-919X
  • Email: ubdarj01@homer.louisville.edu
  • Eric P. Larsen
  • Email: larseep@mail.auburn.edu
  • Received by editor(s): August 20, 1996
  • Received by editor(s) in revised form: May 7, 1997
  • Additional Notes: Work was begun on this paper while the first two authors were participants at the Nineteenth Summer Symposium in Real Analysis, held in Erice, Italy, June 13–20, 1995. The first author acknowledges support from NSF EPSCoR in Alabama, which allowed him to attend this symposium.
  • Communicated by: J. Marshall Ash
  • © Copyright 1999 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 127 (1999), 173-182
  • MSC (1991): Primary 26A48; Secondary 26A24
  • DOI: https://doi.org/10.1090/S0002-9939-99-04571-2
  • MathSciNet review: 1469402