Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-1988-0936773-3
MathSciNet review: 936773
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 the approximate symmetric derivative, Colloq. Math. 28 (1973), 275-285. MR 0327991 (48:6333)
  • [2] L. Larson, Symmetric real analysis: A survey, Real Anal. Exchange 9 (1983-1984), 154-178. MR 742782 (85f:26024)
  • [3] -, Monotonicity and the approximate symmetric derivative, Real Anal. Exchange 12 (1986-1987), 121-123.
  • [4] J. Matousek, Approximate symmetric derivatives and monotonicity, Comment Math. Univ. Carolin. 27 (1986), 83-86. MR 843422 (87f:26004)
  • [5] S. N. Mukhopadhyay, On approximate Schwarz differentiability, Monatsh. Math. 70 (1966), 454-460. MR 0202937 (34:2796)
  • [6] H. W. Pu and H. H. Pu, On approximate Schwarz derivatives, Rev. Roumaine Math. Pures Appl. 25 (1980), 257-264. MR 577036 (81g:26003)

Similar Articles

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

Retrieve articles in all journals with MSC: 26A48, 26A24


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1988-0936773-3
Keywords: Symmetric derivative, approximate symmetric derivative, density, monotonicity
Article copyright: © Copyright 1988 American Mathematical Society

American Mathematical Society