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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Symmetric derivates, scattered, and semi-scattered sets

Author: Chris Freiling
Journal: Trans. Amer. Math. Soc. 318 (1990), 705-720
MSC: Primary 26A24; Secondary 26A48
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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Keywords: Monotonicity, symmetric derivative, scattered sets, symmetric covers, symmetric continuity, symmetric derivation bases
Article copyright: © Copyright 1990 American Mathematical Society

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