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Transactions of the American Mathematical Society

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Symmetric derivates, scattered, and semi-scattered sets


Author: Chris Freiling
Journal: Trans. Amer. Math. Soc. 318 (1990), 705-720
MSC: Primary 26A24; Secondary 26A48
DOI: https://doi.org/10.1090/S0002-9947-1990-0989574-8
MathSciNet review: 989574
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Abstract: We call a set right scattered (left scattered) if every nonempty subset contains a point isolated on the right (left). We establish the following monotonicity theorem for the symmetric derivative. If a real function $ f$ has a nonnegative lower symmetric derivate on an open interval $ I$, then there is a nondecreasing function $ g$ such that $ f(x) > g(x)$ on a right scattered set and $ f(x) < g(x)$ on a left scattered set. Furthermore, if $ R$ is any right scattered set and $ L$ is any left scattered set disjoint with $ R$, then there is a function which is positive on $ R$, negative on $ L$, zero otherwise, and which has a zero lower symmetric derivate everywhere. We obtain some consequence including an analogue of the Mean Value Theorem and a new proof of an old theorem of Charzynski.


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  • [1] C. L. Belna, M. J. Evans, and P. D. Humke, Symmetric monotonicity, Acta Math. Acad. Sci. Hungar. 34 (1979), 17-22. MR 546713 (81b:26005)
  • [2] Z. Charzynski, Sur les fonctions dont la derivee symetrique est partout finie, Fund. Math. 21 (1933), 214-225.
  • [3] M. Chlebik, On symmetrically continuous functions, Real Anal. Exchange 13 (1987-88), 34.
  • [4] Roy O. Davies, Symmetric sets are measurable, Real Anal. Exchange 4 (1978-79), 87-89. MR 525017 (80c:28002)
  • [5] Roy O. Davies and Fred Galvin, Solution to Query $ 5$, Real Anal. Exchange 2 (1976), 74-75.
  • [6] M. J. Evans, A symmetric condition for monotonicity, Bull. Math. Inst. Acad. Sinica 6 (1978), 85-91. MR 0499025 (58:17002)
  • [7] Editorial Staff, Concerning Query $ 37$, Real Anal. Exchange 4 (1978-79), 82-83.
  • [8] J. Foran, The symmetric and ordinary derivative, Real Anal. Exchange 2 (1977), 105-108. MR 700195 (84g:26005)
  • [9] D. Gale and F. M. Stewart, Infinite games of perfect information, Ann. of Math. Stud., no. 28, Princeton Univ. Press, Princeton, N.J., 1953, pp. 245-266. MR 0054922 (14:999b)
  • [10] F. Hausdorff, Problem $ 62$, Fund. Math. 25 (1935), 578.
  • [11] A. Khintchine, Recherches sur la structure des fonctions measurables, Fund. Math. 9 (1927), 212-279.
  • [12] L. Larson, The symmetric derivative, Trans. Amer. Math. Soc. 277 (1983), 589-599. MR 694378 (84j:26009)
  • [13] -, Symmetric real analysis: a survey, Real Anal. Exchange 9 (1983-1984), 154-178. MR 742782 (85f:26024)
  • [14] J. McGrotty, A theorem on complete sets, J. London Math. Soc. 37 (1962), 338-340. MR 0140635 (25:4052)
  • [15] Natanson, Theory of functions of a real variable, Ungar, New York, 1964.
  • [16] D. Preiss, A note on symmetrically continuous functions, Časopis Pěst. Mat. 96 (1971), 262-264. MR 0306411 (46:5537)
  • [17] I. Z. Ruzsa, Locally symmetric functions, Real Anal. Exchange 4 (1978-1979), 84-86.
  • [18] W. Sierpiński, Sur une hypothese de M. Mazurkiewicz, Fund. Math. 11 (1928), 148
  • [19] E. Szpilrajn, Remarque sur la derivee symetrique, Fund. Math. 21 (1931), 226-228.
  • [20] B. S. Thomson, On full covering properties, Real Anal. Exchange 6 (1980-81), 77-93. MR 606543 (82c:26008)
  • [21] J. Uher, Symmetrically differentiable functions are differentiable almost everywhere, Real Anal. Exchange 8 (1982-83), 253-261. MR 694513 (84f:26009)
  • [22] C. E. Weil, Monotonicity, convexity and symmetric derivatives, Trans. Amer. Math. Soc. 222 (1976), 225-237. MR 0401994 (53:5817)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1990-0989574-8
Keywords: Monotonicity, symmetric derivative, scattered sets, symmetric covers, symmetric continuity, symmetric derivation bases
Article copyright: © Copyright 1990 American Mathematical Society

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