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

DOI:
https://doi.org/10.1090/S0002-9939-99-04571-2

MathSciNet review:
1469402

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Abstract | References | Similar Articles | Additional Information

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 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:
https://doi.org/10.1090/S0002-9939-99-04571-2

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