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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
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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
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.
Formally, a modulus of continuity is any increasing real-extended valued function ω : [0, ∞] → [0, ∞], vanishing at 0 and continuous at 0, that is
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,
Also, f admits ω as (global) modulus of continuity if and only if,
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.
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 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. This is always true in case of either compact or convex domains. 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. 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′) with distance bounded away from zero; this condition is certainly satisfied by any bounded uniformly continuous function; hence in particular, by any continuous function on a compact metric space.
A sublinear modulus of continuity can easily be found for any uniformly continuous function which is a bounded perturbation of a Lipschitz function: if f is a uniformly continuous function with modulus of continuity ω, and g is a k Lipschitz function with uniform distance r from f, then f admits the sublinear module of continuity min{ω(t), 2r+kt}. Conversely, at least for real-valued functions, any special uniformly continuous function is a bounded, uniformly continuous perturbation of some Lipschitz function; indeed more is true as shown below (Lipschitz approximation).
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. Therefore, f∗ and f* are respectively inferior and superior envelopes of ω-continuous families; hence still ω-continuous. Incidentally, by the Kuratowski embedding any metric space is isometric to a subset of a normed space. Hence, special uniformly continuous real-valued functions are essentially the restrictions of uniformly continuous functions on normed spaces. 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.
Every special uniformly continuous real-valued function f : X → R defined on the metric space X is uniformly approximable by means of Lipschitz functions. 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. Correspondingly, an optimal approximation is given by the functions
each function fs has Lipschitz constant s and
in fact, it is the greatest s-Lipschitz function that realize the distance δ(s). For example, the α-Hölder real-valued functions on a metric space are characterized as those functions that can be uniformly approximated by s-Lipschitz functions with speed of convergence while the almost Lipschitz functions are characterized by an exponential speed of convergence
Steffens (2006, p. 160) attributes the first usage of omega for the modulus of continuity to Lebesgue (1909, p. 309/p. 75) where omega refers to the oscillation of a Fourier transform. De la Vallée Poussin (1919, pp. 7-8) mentions both names (1) "modulus of continuity" and (2) "modulus of oscillation" and then concludes "but we choose (1) to draw attention to the usage we will make of it".
Let 1 ≤ p; let f : Rn → R a function of class Lp, and let h ∈ Rn. 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. In the case p = ∞ the above property does not hold in general: actually, it exactly reduces to the uniform continuity, and defines the uniform continuous functions. This leads to the following definition, that generalizes the notion of a modulus of continuity of the uniformly continuous functions: a modulus of continuity Lp for a measurable function f : X → R is a modulus of continuity ω : [0, ∞] → [0, ∞] such that
This way, moduli of continuity also give a quantitative account of the continuity property shared by all Lp functions.
It can be seen that formal definition of the modulus uses notion of finite difference of first order:
If we replace that difference with a difference of order n, we get a modulus of continuity of order n:
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