The following problem posed by G. L. Itzkowitz is answered (in affirmative) in the paper: Are the right and left uniformities of a locally compact group equivalent iff every real valued left uniformly continuous function is right uniformly continuous? Moreover, it is proved that the coincidence of the left and right uniformly continuous real valued functions implies the coincidence of the left and right uniformities of an almost metrizable group (in the sense of B. A. Pasynkov). There is not any known counterexample in general. Remark: Recently the original problem was also settled by \textit{G. L. Itzkowitz} [Uniformities and Uniform Continuity on Topological Groups, Proc. Summer Top. Conf. Staten Island, preprint].