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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A symmetric density property: monotonicity and the approximate symmetric derivative

Authors: C. Freiling and D. Rinne
Journal: Proc. Amer. Math. Soc. 104 (1988), 1098-1102
MSC: Primary 26A48; Secondary 26A24
MathSciNet review: 936773
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Abstract: The following is established:

Let $ W$ and $ B$ be open sets of real numbers whose union has full measure. If for each $ x$, the set $ \{ h > 0\vert x - h \in W,x + h \in B\} $ has density zero at zero, then these sets are all empty.

This is then used to prove the following:

If $ f$ is a continuous real valued function with a nonnegative lower approximate symmetric derivative, then $ f$ is nondecreasing.

References [Enhancements On Off] (What's this?)

  • [1] N. K. Kundu, On approximate symmetric derivative, Colloq. Math. 28 (1973), 275–285. MR 0327991
  • [2] Lee Larson, Symmetric real analysis: a survey, Real Anal. Exchange 9 (1983/84), no. 1, 154–178. MR 742782
  • [3] -, Monotonicity and the approximate symmetric derivative, Real Anal. Exchange 12 (1986-1987), 121-123.
  • [4] Jiří Matoušek, Approximate symmetric derivative and monotonicity, Comment. Math. Univ. Carolin. 27 (1986), no. 1, 83–86. MR 843422
  • [5] S. N. Mukhopadhyay, On approximate Schwarz differentiability, Monatsh. Math. 70 (1966), 454–460. MR 0202937
  • [6] H. H. Pu and H. W. Pu, On approximate Schwarz derivates, Rev. Roumaine Math. Pures Appl. 25 (1980), no. 2, 257–264. MR 577036

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Keywords: Symmetric derivative, approximate symmetric derivative, density, monotonicity
Article copyright: © Copyright 1988 American Mathematical Society