Modulus of continuity and Lipschitz approximation

Given a uniformly continuous map f from a doubling metric space X to a normed linear space V, and given a subadditive function ω, we give a characterization of ω dominating the modulus of continuity of f in terms of Lipschitz approximation. As the main part of this characterization, we give a constructive method of approximating a uniformly continuous map from X to V by Lipschitz maps, the corresponding approximation operation is linear.