Page "Modulus of continuity" Paragraph 5

A special role is played by concave moduli of continuity, especially in connection with extension properties, and with approximation of uniformly continuous functions.

For a function between metric spaces, it is equivalent to admit a modulus of continuity that is either concave, or subadditive, or uniformly continuous, or sublinear ( in the sense of growth ).

Actually, the existence of such special moduli of continuity for a uniformly continuous function is always ensured whenever the domain is either a compact, or a convex subset of a normed space.

However, a uniformly continuous function on a general metric space admits a concave modulus of continuity if and only if the ratios are uniformly bounded for all pairs bounded away from the diagonal of.

The functions with the latter property constitute a special subclass of the uniformly continuous functions, that in the following we refer to as the special uniformly continuous functions.