IJPAM: Volume 78, No. 1 (2012)

POINTS OF SYMMETRIC CONTINUITY
OF REAL FUNCTIONS

Kandasamy Muthuvel
Department of Mathematics
University of Wisconsin-Oshkosh
Oshkosh, Wisconsin 54901-8601, USA


Abstract. We examine the set of points where a real function is symmetric or symmetrically continuous but not continuous. Among other things, we show that if $G$ is a proper additive subgroup of the reals, then there exists a real function $f$ with two-element range such that the set of points where $f$ is symmetrically continuous but not continuous is the additive subgroup $G$. The above statement is not true if symmetrically continuous is replaced by symmetric. However, there exists a real function $f$ with three-element range such that the set of points where $f$ is symmetric but not continuous is the additive subgroup $G$. In both results, $G$ can not be replaced by $\mathbb{R}$.

Received: December 30, 2011

AMS Subject Classification: 26A15, 26A18

Key Words and Phrases: symmetric functions, symmetrically continuous functions, residual set, arithmetic progression

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2012
Volume: 78
Issue: 1