In previous papers, we
discussed the extension of uniformly continuous real-valued mappings from subspaces
of metric spaces and the same question for mappings into certain Banach spaces, such
as c0(I) and l∞(I). Since the extension of uniformly continuous mappings into l∞(I)
is equivalent to the extension of equi-uniformly continuous point bounded families of
real-valued mappings, it is natural to ask about the extension of equi-uniformly
continuous families which are not necessarily point-bounded. The present paper
investigates this extension property and several related questions concerning the
extension of uniformly continuous mappings with values in uniformly discrete
spaces.
