Modulus of continuity

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 and only if

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 equicontinuous families. For instance, the modulus ω(t) := kt describes the k-Lipschitz functions, the moduli ω(t) := kt 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 uniform continuity. The same notions generalize naturally to functions between metric spaces. Moreover, a suitable local version of these notions allows to describe quantitatively the continuity at a point in terms of moduli of continuity.