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 $Lip^1$.References
- Andrew Bruckner, Differentiation of real functions, 2nd ed., CRM Monograph Series, vol. 5, American Mathematical Society, Providence, RI, 1994. MR 1274044, DOI 10.1090/crmm/005
- 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 476939, DOI 10.1090/S0002-9947-1977-0476939-X
- U. B. Darji, Two general extension theorems, submitted.
- K. M. Garg, On nowhere monotone functions. I. Derivatives at a residual set, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 5 (1962), 173–177. MR 146317
- K. M. Garg, 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 177075
- K. M. Garg, On nowhere monotone functions. III. (Functions of first and second species), Rev. Math. Pures Appl. 8 (1963), 83–90. MR 151560
- K. M. Garg, On level sets of a continuous nowhere monotone function, Fund. Math. 52 (1963), 59–68. MR 143855, DOI 10.4064/fm-52-1-59-68
- K. M. Garg, Construction of absolutely continuous and singular functions that are nowhere of monotonic type, Classical real analysis (Madison, Wis., 1982) Contemp. Math., vol. 42, Amer. Math. Soc., Providence, RI, 1985, pp. 61–79. MR 807979, DOI 10.1090/conm/042/807979
- K. Kuratowski, Topology. Vol. I, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1966. New edition, revised and augmented; Translated from the French by J. Jaworowski. MR 0217751
- S. Marcus, Sur les fonctions continues qui ne sont monotones en aucun intervalle, Rev. Math. Pures Appl. 3 (1958), 101–105 (French). MR 107680
- John C. Oxtoby, Measure and category. A survey of the analogies between topological and measure spaces, Graduate Texts in Mathematics, Vol. 2, Springer-Verlag, New York-Berlin, 1971. MR 0393403
- Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82–96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
- G. Petruska and M. Laczkovich, Baire $1$ functions, approximately continuous functions and derivatives, Acta Math. Acad. Sci. Hungar. 25 (1974), 189–212. MR 379766, DOI 10.1007/BF01901760
- S. Saks, Theory of the Integral, Monografie Mat., Vol. 7, PWN, Warsaw, 1937.
- W. Sierpiński, Sur l’ensemble de valeurs qu’une fonction continue prend une infinité non dénombrable de fois, Fund. Math. 8 (1926), 370-373.
- A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776
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