Uniformly antisymmetric function with bounded range

The goal of this note is to construct a uniformly antisymmetric function f:R-> R with a bounded countable range. This answers Problem 1(b) of Ciesielski and Larson. (See also list of problems in Thomson and Problem 2(b) from Ciesielskis survey.) A problem of existence of uniformly antisymmetric function f:R-> R with finite range remains open.