# IJPAM: Volume 78, No. 1 (2012)

**POINTS OF SYMMETRIC CONTINUITY**

OF REAL FUNCTIONS

OF REAL FUNCTIONS

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 is a proper additive subgroup of the reals, then there exists a real function with two-element range such that the set of points where is symmetrically continuous but not continuous is the additive subgroup . The above statement is not true if symmetrically continuous is replaced by symmetric. However, there exists a real function with three-element range such that the set of points where is symmetric but not continuous is the additive subgroup . In both results, can not be replaced by .

**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