In this note it is proved that the range of uniformly antisymmetric function must have at least four elements. This generalizes the results from [P. Kostyrko, There is no strongly locally antisymmetric set, Real Analysis Exchange 17 (1991-92), 423--425] and [K. Ciesielski, L. Larson, Uniformly antisymmetric functions, Real Anal. Exchange 19 (1993-94), 226-235] that the range of such function must have at least three elements. The problem whether the range of uniformly antisymmetric function can be finite remains open.
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Last modified July 5, 2001.