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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