Editing Modulus of continuity (section)

{{Short description|Function in mathematical analysis}}
{{accuracy|The definition given does not match the one common in the literature?|date=November 2023}}
In [[mathematical analysis]], a '''modulus of continuity''' is a function ω : [0, ∞] → [0, ∞] used to measure quantitatively the [[uniform continuity]] of functions. So, a function ''f'' : ''I'' → '''R''' admits ω as a modulus of continuity if
:<math>|f(x)-f(y)|\leq\omega(|x-y|),</math>
for all ''x'' and ''y'' in the domain of ''f''. Since moduli of continuity are required to be infinitesimal at 0, a function turns out to be uniformly continuous if and only if it admits a modulus of continuity. Moreover, relevance to the notion is given by the fact that sets of functions sharing the same modulus of continuity are exactly [[equicontinuity|equicontinuous families]]. For instance, the modulus ω(''t'') := ''kt'' describes the k-[[Lipschitz functions]], the moduli ω(''t'') := ''kt''<sup>α</sup> describe the [[Hölder continuity]], the modulus ω(''t'') := ''kt''(|log ''t''|+1) describes the '''almost Lipschitz''' class, and so on. In general, the role of ω is to fix some explicit functional dependence of ε on δ in the [[(ε, δ)-definition of limit#Uniform continuity|(ε, δ) definition of uniform continuity]]. The same notions generalize naturally to functions between [[metric space]]s. Moreover, a suitable local version of these notions allows to describe quantitatively the continuity at a point in terms of moduli of continuity.

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 [[linear growth|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

:<math>\frac{d_Y(f(x),f(x'))}{d_X(x,x')}</math>

are uniformly bounded for all pairs (''x'', ''x''′) bounded away from the diagonal of ''X x X''. 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. Real-valued special uniformly continuous functions on the metric space ''X'' can also be characterized as the set of all functions that are restrictions to ''X'' of uniformly continuous functions over any normed space isometrically containing ''X''. Also, it can be characterized as the uniform closure of the Lipschitz functions on ''X''.