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
| f ( x ) − f ( y ) | ≤ ω ( | x − y | ) , {\displaystyle |f(x)-f(y)|\leq \omega (|x-y|),}
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) := ktα 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. 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
d Y ( f ( x ) , f ( x ′ ) ) d X ( x , x ′ ) {\displaystyle {\frac {d_{Y}(f(x),f(x'))}{d_{X}(x,x')}}}
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.
Formal definition Formally, a modulus of continuity is any increasing real-extended valued function ω : [0, ∞] → [0, ∞], vanishing at 0 and continuous at 0, that is
lim t → 0 ω ( t ) = ω ( 0 ) = 0. {\displaystyle \lim _{t\to 0}\omega (t)=\omega (0)=0.}
Moduli of continuity are mainly used to give a quantitative account both of the continuity at a point, and of the uniform continuity, for functions between metric spaces, according to the following definitions. A function f : (X, dX) → (Y, dY) admits ω as (local) modulus of continuity at the point x in X if and only if,
∀ x ′ ∈ X : d Y ( f ( x ) , f ( x ′ ) ) ≤ ω ( d X ( x , x ′ ) ) . {\displaystyle \forall x'\in X:d_{Y}(f(x),f(x'))\leq \omega (d_{X}(x,x')).}
Also, f admits ω as (global) modulus of continuity if and only if,
∀ x , x ′ ∈ X : d Y ( f ( x ) , f ( x ′ ) ) ≤ ω ( d X ( x , x ′ ) ) . {\displaystyle \forall x,x'\in X:d_{Y}(f(x),f(x'))\leq \omega (d_{X}(x,x')).}
One equivalently says that ω is a modulus of continuity (resp., at x) for f, or shortly, f is ω-continuous (resp., at x). Here, we mainly treat the global notion.
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