Moreover, relevance to the notion is given by the fact that sets of functions sharing the same modulus of continuity are exactly equicontinuous families.
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 (x, x′) bounded away from the diagonal of X x X.
Special moduli of continuity also reflect certain global properties of functions such as extendibility and uniform approximation.
These properties are essentially equivalent in that, for a modulus ω (more precisely, its restriction on [0, ∞)) each of the following implies the next: Thus, for a function f 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 f is sometimes called a special uniformly continuous map.
Indeed, a uniformly continuous map f : C → Y defined on a convex set C of a normed space E always admits a subadditive modulus of continuity; in particular, real-valued as a function ω : [0, ∞) → [0, ∞).
Indeed, it is immediate to check that the optimal modulus of continuity ωf defined above is subadditive if the domain of f is convex: we have, for all s and t: 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 a and b such that |f(x)| ≤ a|x|+b for all x.
The above property for uniformly continuous function on convex domains admits a sort of converse at least in the case of real-valued functions: that is, every special uniformly continuous real-valued function f : X → R defined on a metric space X, which is a metric subspace of a normed space E, admits extensions over E that preserves any subadditive modulus ω of f. The least and the greatest of such extensions are respectively: As remarked, any subadditive modulus of continuity is uniformly continuous: in fact, it admits itself as a modulus of continuity.
In particular, this construction provides a quick proof of the Tietze extension theorem on compact metric spaces.
However, for mappings with values in more general Banach spaces than R, the situation is quite more complicated; the first non-trivial result in this direction is the Kirszbraun theorem.
Moreover, the speed of convergence in terms of the Lipschitz constants of the approximations is strictly related to the modulus of continuity of f. Precisely, let ω be the minimal concave modulus of continuity of f, which is Let δ(s) be the uniform distance between the function f and the set Lips of all Lipschitz real-valued functions on C having Lipschitz constant s : Then the functions ω(t) and δ(s) can be related with each other via a Legendre transformation: more precisely, the functions 2δ(s) and −ω(−t) (suitably extended to +∞ outside their domains of finiteness) are a pair of conjugated convex functions,[1] for Since ω(t) = o(1) for t → 0+, it follows that δ(s) = o(1) for s → +∞, that exactly means that f is uniformly approximable by Lipschitz functions.
Steffens (2006, p. 160) attributes the first usage of omega for the modulus of continuity to Lebesgue (1909, p. 309/p.
The h-translation of f, the function defined by (τhf)(x) := f(x−h), belongs to the Lp class; moreover, if 1 ≤ p < ∞, then as ǁhǁ → 0 we have: Therefore, since translations are in fact linear isometries, also as ǁhǁ → 0, uniformly on v ∈ Rn.
In other words, the map h → τh defines a strongly continuous group of linear isometries of Lp.