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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
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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.


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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

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