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

 

Nowhere monotone functions
and functions of nonmonotonic type


Authors: Jack B. Brown, Udayan B. Darji and Eric P. Larsen
Journal: Proc. Amer. Math. Soc. 127 (1999), 173-182
MSC (1991): Primary 26A48; Secondary 26A24
MathSciNet review: 1469402
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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 ${\cal P}^n,\ C^n$, 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
Email: ubdarj01@homer.louisville.edu

Eric P. Larsen
Email: larseep@mail.auburn.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-99-04571-2
PII: S 0002-9939(99)04571-2
Keywords: Nowhere monotone, nonmonotonic type, level sets
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
Article copyright: © Copyright 1999 American Mathematical Society