In this paper we use an
observation of M. Rajagopalan to show that each nondiscrete locally compact
topological group can be wlitten as a disjoint union of continuumly many closed
nowhere dense Gδ sets. This observation also enables us to give a new constructive
proof of a theorem of Kister. We show that in a nondiscrete noncompact locally
compact group it is always possible to construct a bounded continuous function that
is not left uniformly continuous. Finally this construction motivates a similar
construction which yields examples of functions in LUC(G) but not in RUC (G)
when G is a nondiscrete and nonunimodular locally compact topological group or
when G is a nondiscrete locally compact metric group with inequivalent right and left
uniform stmctures.
