Proceedings of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6826(e) ISSN 0002-9939(p)

Nowhere monotone functions and functions of nonmonotonic type

Author(s): Jack B. Brown; Udayan B. Darji; Eric P. Larsen
Journal: Proc. Amer. Math. Soc. 127 (1999), 173-182.
MSC (1991): Primary 26A48; Secondary 26A24
MathSciNet review: 1469402
Retrieve article in: PDF
This article is available free of charge

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 ${\cal P}^n, C^n$ , and .

References:

1.: A. M. Bruckner, Differentiation of Real Functions, CRM Monograph Series Vol. 5, Amer. Math. Soc., Providence, 1994. MR 94m:26001
2.: A. M. Bruckner and K. M. Garg, The level structure of a residual set of continuous functions, Trans. Amer. Math. Soc. 232 (1977), 307-321. MR 57:16487
3.: U. B. Darji, Two general extension theorems, submitted.
4.: K. M. Garg, On nowhere monotone functions. I. Derivates at residual set, Ann. Univ. Sci. Budapest, Eötvös Sect. Math. 5 (1962), 173-177. MR 26:3839
5.: -, On nowhere monotone functions. II. Derivates at sets of power c and at sets of positive measure, Rev. Math. Pures Appl. 7 (1962) 663-671. MR 31:1339
6.: -, On nowhere monotone functions. III. (Functions of first and second species), Rev. Math. Pures Appl. 8 (1963) 83-90. MR 27:1545
7.: -, On level sets of a continuous nowhere monotone function, Fund. Math. 52 (1963), 59-68. MR 26:1405
8.: -, Construction of absolutely continuous and singular functions that are nowhere of monotonic type, Contemp. Math. 42 (1985), 61-79. MR 87d:26008
9.: C. Kuratowski, Topology. I and II, Academic Press, New York 1966, 1968. MR 36:840; MR 41:4467
10.: S. Marcus, Sur les fonctions continues qui ne sont monotones en aucun intervalle, Rev. Math. Pures Appl. 3 (1958), 101-105. MR 21:6403
11.: J. Oxtoby, Measure and Category, Springer-Verlag, New York 1971. MR 52:14213
12.: K. Padmavally, On the roots of equation $f(x) = \xi$ where is real and continuous in but monotonic in no subinterval of , Proc. Amer. Math. Soc. 4 (1953), 839-841. MR 15:513h
13.: G. Petruska and M. Laczkovich, Baire 1 functions, approximately continuous functions, and derivatives, Acta. Math. Hungar. 25 (1974), 189-212. MR 52:671
14.: S. Saks, Theory of the Integral, Monografie Mat., Vol. 7, PWN, Warsaw, 1937.
15.: W. Sierpi\'{n}ski, Sur l'ensemble de valeurs qu'une fonction continue prend une infinité non dénombrable de fois, Fund. Math. 8 (1926), 370-373.
16.: A. Zygmund, Trigonometric Series, Vol. I, Cambridge Univ. Press, London, 1959. MR 21:6498

Similar Articles:

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 26A48, 26A24

Retrieve articles in all Journals with MSC (1991): 26A48, 26A24

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
Affiliation: Department of Mathematics, Auburn University, Auburn, Alabama 36849-5310
Email: larseep@mail.auburn.edu

DOI: 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 \hbox{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 of article: Copyright 1999, American Mathematical Society

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|