Modulus of Continuity - Special Moduli of Continuity

Special Moduli of Continuity

Special moduli of continuity also reflect certain global properties of functions such as extendibility and uniform approximation. In this section we mainly deal with moduli of continuity that are concave, or subadditive, or uniformly continuous, or sublinear. These properties are essentially equivalent in that, for a modulus (more precisely, its restriction on ) each of the following implies the next:

• is concave;
• is subadditive;
• is uniformly continuous;
• is sublinear, that is, there are constants and such that for all ;
• is dominated by a concave modulus, that is, there exists a concave modulus of continuity such that for all .

Thus, for a function between metric spaces it is equivalent to admit a modulus of continuity which is either concave, or subadditive, or uniformly continuous, or sublinear. In this case, the function is sometimes called a special uniformly continuous map. This is always true in case of either compact or convex domains. Indeed, a uniformly continuous map defined on a convex set of a normed space always admits a subadditive modulus of continuity; in particular, real-valued as a function . Indeed, it is immediate to check that the optimal modulus of continuity defined above is subadditive if the domain of is convex: we have, for all and :

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 ; this condition is certainly satisfied by any bounded uniformly continuous function; hence in particular, by any continuous function on a compact metric space.