Modulus of continuity

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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 : IR 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) := 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. 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 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.

Elementary facts

Remarks

  • Some authors require additional properties such as ω being increasing, or continuous. However, if f admits a modulus of continuity in the weaker definition above, it also admits a modulus of continuity which is increasing and infinitely differentiable in ]0, ∞[. For instance,
is increasing, and ω1 ≥ ω;
is also continuous, and ω2 ≥ ω1,
and a suitable variant of the preceding definition also makes ω2 infinitely differentiable in ]0, ∞[.
  • Any uniformly continuous function admits a minimal modulus of continuity ωf, that is sometimes referred to as the (optimal) modulus of continuity of f:
Similarly, any function continuous at the point x admits a minimal modulus of continuity at x, ωf(t; x) (the (optimal) modulus of continuity of f at x) :
However, these restricted notions are not as relevant, for in most cases the optimal modulus of f could not be computed explicitly, but only bounded from above (by any modulus of continuity of f). Moreover, the main properties of moduli of continuity concern directly the unrestricted definition.
  • In general, the modulus of continuity of a uniformly continuous function on a metric space needs to take the value +∞. For instance, the function f : NN such that f(n) := n2 is uniformly continuous with respect to the discrete metric on N, and its minimal modulus of continuity is ωf(t) = +∞ for any positive integer t, and ωf(t) = 0 otherwise. However, the situation is different for uniformly continuous functions defined on compact or convex subsets of normed spaces.

Special moduli of continuity

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 : CY 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:

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; this condition is certainly satisfied by any bounded uniformly continuous function; hence in particular, by any continuous function on a compact metric space.

Sublinear moduli, and bounded perturbations from Lipschitz

A sublinear modulus of continuity can easily found for any uniformly function which is a bounded perturbations 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 bounded, uniformly continuous perturbation of a Lipschitz function is a special uniformly continuous function; indeed more is true as shown below. 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.

Subadditive moduli, and extendibility

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 : XR defined on a subset X 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.

Concave moduli, and Lipschitz approximation

Every special uniformly continuous real-valued function f : XR 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, 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

Examples of use

  • For an example of use in the Fourier series, see Dini test.

History

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".

The translation group of Lp functions, and moduli of continuity Lp.

Let 1 ≤ p; let f : RnR a function of class Lp, and let hRn. The h-translation of f, the function defined by (τhf)(x) := f(xh), 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 vRn.

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 : XR 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.

Modulus of continuity of higher orders

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:

See also

References

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