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A sublinear modulus of continuity can easily found for any uniformly function which is a bounded perturbations of a Lipschitz function: if is a uniformly continuous function with modulus of continuity, and is a Lipschitz function with uniform distance from, then admits the sublinear module of continuity Conversely, at least for real-valued functions, any bounded, uniformly continuous perturbation of a Lipschitz function is a special uniformly continuous function ; indeed more is true as shown below.
Note that as an immediate consequence, any uniformly continuous function on a convex subset of a normed space has a sublinear growth: there are constants and such that for all.

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