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), 173182
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 , and .
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Additional Information
Jack B. Brown
Affiliation:
Department of Mathematics, Auburn University, Auburn, Alabama 368495310
Email:
brownj4@mail.auburn.edu
Udayan B. Darji
Affiliation:
Department of Mathematics, University of Louisville, Louisville, Kentucky 402920001
Email:
ubdarj01@homer.louisville.edu
Eric P. Larsen
Email:
larseep@mail.auburn.edu
DOI:
http://dx.doi.org/10.1090/S0002993999045712
PII:
S 00029939(99)045712
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
