modulus of continuity
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mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
, a modulus of continuity is a function ω : , ∞, ∞used to measure quantitatively the
uniform continuity In mathematics, a real function f of real numbers is said to be uniformly continuous if there is a positive real number \delta such that function values over any function domain interval of the size \delta are as close to each other as we want. In ...
of functions. So, a function ''f'' : ''I'' → R admits ω as a modulus of continuity if and only if :, 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 In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there ...
, the moduli ω(''t'') := ''kt''α describe the
Hölder continuity Hölder: * ''Hölder, Hoelder'' as surname * Hölder condition * Hölder's inequality * Hölder mean * Jordan–Hölder theorem In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a modu ...
, 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 space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
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
growth Growth may refer to: Biology * Auxology, the study of all aspects of human physical growth * Bacterial growth * Cell growth * Growth hormone, a peptide hormone that stimulates growth * Human development (biology) * Plant growth * Secondary 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 :\frac 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 : , ∞, ∞ vanishing at 0 and continuous at 0, that is :\lim_\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, :\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, :\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.


Elementary facts

*If ''f'' has ω as modulus of continuity and ω1 ≥ ω, then ''f'' admits ω1 too as modulus of continuity. *If ''f'' : ''X'' → ''Y'' and ''g'' : ''Y'' → ''Z'' are functions between metric spaces with moduli respectively ω1 and ω2 then the composition map g\circ f:X\to Z has modulus of continuity \omega_2\circ\omega_1. *If ''f'' and ''g'' are functions from the metric space X to the Banach space ''Y'', with moduli respectively ω1 and ω2, then any linear combination ''af''+''bg'' has modulus of continuity , ''a'', ω1+, ''b'', ω2. In particular, the set of all functions from ''X'' to ''Y'' that have ω as a modulus of continuity is a convex subset of the vector space ''C''(''X'', ''Y''), closed under
pointwise convergence In mathematics, pointwise convergence is one of Modes of convergence (annotated index), various senses in which a sequence of functions can Limit (mathematics), converge to a particular function. It is weaker than uniform convergence, to which it i ...
. *If ''f'' and ''g'' are bounded real-valued functions on the metric space ''X'', with moduli respectively ω1 and ω2, then the pointwise product ''fg'' has modulus of continuity \, g\, _\infty\omega_1+\, f\, _\infty \omega_2. *If \_ is a family of real-valued functions on the metric space ''X'' with common modulus of continuity ω, then the inferior envelope \inf_f_\lambda, respectively, the superior envelope \sup_f_\lambda, is a real-valued function with modulus of continuity ω, provided it is finite valued at every point. If ω is real-valued, it is sufficient that the envelope be finite at one point of ''X'' at least.


Remarks

*Some authors do not require monotonicity, and some require additional properties such as ω being continuous. However, if f admits a modulus of continuity in the weaker definition, it also admits a modulus of continuity which is increasing and infinitely differentiable in ]0, ∞[. For instance, \omega_1(t) := \sup_\omega(s) is increasing, and ω1 ≥ ω; \omega_2(t):=\frac \int_t^\omega_1(s)ds 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 ω''f'', that is sometimes referred to as ''the'' (optimal) modulus of continuity of ''f'': \omega_f(t) := \sup\ ,\quad\forall t\geq0. Similarly, any function continuous at the point ''x'' admits a minimal modulus of continuity at ''x'', ω''f''(''t''; ''x'') (''the'' (optimal) modulus of continuity of ''f'' at ''x'') : \omega_f(t;x):=\sup\,\quad\forall t\geq0. 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'' : N → N such that ''f''(''n'') := ''n''2 is uniformly continuous with respect to the
discrete metric Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory * Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit *Discrete group, a ...
on N, and its minimal modulus of continuity is ω''f''(''t'') = +∞ for any ''t''≥1, and ω''f''(''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 Concave or concavity may refer to: Science and technology * Concave lens * Concave mirror Mathematics * Concave function, the negative of a convex function * Concave polygon, a polygon which is not convex * Concave set * The concavity In ca ...
, or
subadditive In mathematics, subadditivity is a property of a function that states, roughly, that evaluating the function for the sum of two elements of the domain always returns something less than or equal to the sum of the function's values at each element. ...
, 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: *ω is concave; *ω is subadditive; *ω is uniformly continuous; *ω is sublinear, that is, there are constants ''a'' and ''b'' such that ω(''t'') ≤ ''at''+''b'' for all ''t''; *ω is dominated by a concave modulus, that is, there exists a concave modulus of continuity \tilde\omega such that \omega(t)\leq \tilde\omega(t) for all ''t''. 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 In mathematics, subadditivity is a property of a function that states, roughly, that evaluating the function for the sum of two elements of the domain always returns something less than or equal to the sum of the function's values at each element. ...
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 ω''f'' defined above is subadditive if the domain of ''f'' is convex: we have, for all ''s'' and ''t'': :\begin \omega_f(s+t) &=\sup_ d_Y(f(x),f(x')) \\ &\leq \sup_\left\ \\ &\leq \omega_f(t)+\omega_f(s). \end 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 d_Y(f(x),f(x'))/d_X(x,x') 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.


Sublinear moduli, and bounded perturbations from Lipschitz

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. 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).


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'' : ''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: :\begin f_*(x) &:=\sup_\left\, \\ f^*(x) &:=\inf_\left\. \end 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 In topology, the Tietze extension theorem (also known as the Tietze–Urysohn–Brouwer extension theorem) states that continuous functions on a closed subset of a normal topological space can be extended to the entire space, preserving boundedness ...
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 In mathematics, specifically real analysis and functional analysis, the Kirszbraun theorem states that if is a subset of some Hilbert space , and is another Hilbert space, and : f: U \rightarrow H_2 is a Lipschitz-continuous map, then there ...
.


Concave moduli and Lipschitz approximation

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 :\omega(t)=\inf\big\. Let δ(''s'') be the uniform
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between the function ''f'' and the set Lip''s'' of all Lipschitz real-valued functions on ''C'' having Lipschitz constant ''s'' : :\delta(s):=\inf\big\\leq+\infty. Then the functions ω(''t'') and δ(''s'') can be related with each other via a
Legendre transformation In mathematics, the Legendre transformation (or Legendre transform), named after Adrien-Marie Legendre, is an involutive transformation on real-valued convex functions of one real variable. In physical problems, it is used to convert functions of ...
: more precisely, the functions 2δ(''s'') and −ω(−''t'') (suitably extended to +∞ outside their domains of finiteness) are a pair of conjugated convex functions,Legendre transform and Lipschitz approximation
/ref> for :2\delta(s)=\sup_\left\, :\omega(t)=\inf_\left\. 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 :f_s:=\delta(s)+\inf_\, \quad \mathrm \ s\in\mathrm(\delta): each function ''fs'' has Lipschitz constant ''s'' and :\, f-f_s\, _=\delta(s); 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 O(s^), while the almost Lipschitz functions are characterized by an exponential speed of convergence O(e^).


Examples of use

*Let ''f'' : 'a'', ''b''→ R a continuous function. In the proof that ''f'' is
Riemann integrable In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of Göt ...
, one usually bounds the distance between the upper and lower
Riemann sums In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum. It is named after nineteenth century German mathematician Bernhard Riemann. One very common application is approximating the area of functions or ...
with respect to the Riemann partition ''P'' := in terms of the modulus of continuity of ''f'' and the
mesh A mesh is a barrier made of connected strands of metal, fiber, or other flexible or ductile materials. A mesh is similar to a web or a net in that it has many attached or woven strands. Types * A plastic mesh may be extruded, oriented, ex ...
of the partition ''P'' (which is the number , P, := \max_ (t_-t_i) ) S^*(f;P) - S_*(f;P) \leq (b-a) \omega(, P, ). *For an example of use in the Fourier series, see
Dini test In mathematics, the Dini and Dini–Lipschitz tests are highly precise tests that can be used to prove that the Fourier series of a function converges at a given point. These tests are named after Ulisse Dini and Rudolf Lipschitz. Definition L ...
.


History

Steffens (2006, p. 160) attributes the first usage of omega for the modulus of continuity to
Lebesgue Henri Léon Lebesgue (; June 28, 1875 – July 26, 1941) was a French mathematician known for his theory of integration, which was a generalization of the 17th-century concept of integration—summing the area between an axis and the curve of ...
(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'' : R''n'' → R a function of class ''Lp'', and let ''h'' ∈ R''n''. The ''h''-
translation Translation is the communication of the Meaning (linguistic), meaning of a #Source and target languages, source-language text by means of an Dynamic and formal equivalence, equivalent #Source and target languages, target-language text. The ...
of ''f'', the function defined by (τ''h''''f'')(''x'') := ''f''(''x''−''h''), belongs to the ''Lp'' class; moreover, if 1 ≤ ''p'' < ∞, then as ǁ''h''ǁ → 0 we have: :\, \tau_h f - f\, _p=o(1). Therefore, since translations are in fact linear isometries, also :\, \tau_ f - \tau_v f\, _p=o(1), as ǁ''h''ǁ → 0, uniformly on ''v'' ∈ R''n''. In other words, the map ''h'' → τ''h'' 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 : , ∞, ∞such that :\, \tau_h f - f\, _p\leq \omega(h). 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 A finite difference is a mathematical expression of the form . If a finite difference is divided by , one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for t ...
of first order: :\omega_f(\delta)=\omega(f, \delta)=\sup\limits_\left, \Delta_h(f,x)\. If we replace that difference with a difference of order ''n'', we get a modulus of continuity of order ''n'': :\omega_n(f, \delta)=\sup\limits_\left, \Delta^n_h(f,x)\.


See also

*
Constructive analysis In mathematics, constructive analysis is mathematical analysis done according to some principles of constructive mathematics. This contrasts with ''classical analysis'', which (in this context) simply means analysis done according to the (more comm ...
*
Modulus of convergence In real analysis, a branch of mathematics, a modulus of convergence is a function that tells how quickly a convergent sequence converges. These moduli are often employed in the study of computable analysis and constructive mathematics. If a sequ ...


References

* * * Reproduced in: * * *{{cite book , first=K.-G. , last=Steffens , title=The History of Approximation Theory , publisher=Birkhäuser , location=Boston , year=2006 , isbn=0-8176-4353-2 Lipschitz maps Approximation theory Constructivism (mathematics) Fourier analysis