Remote Access Transactions of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9947-1990-0989574-8
MathSciNet review: 989574
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


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


Similar Articles

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

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


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