We study conditions under which universally measurable mappings from a separable topological space S into a metric space R (with metric ρ) belong to class D of mappings f : S → R : such that for any compact subset K ⊂ S and number ε > 0 there exists an open (in the induced topology) set V ⊂ K such that the oscillation ω(f;V) of an R-valued function f on V is less than ε (here, ω(f;V) = sup_{s, t∈V}ρ(f(s), f(t))).