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Modulus of continuity

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In <a href="Mathematical_analysis" title="Mathematical analysis">mathematical analysis</a>, a modulus of continuity is a function ω : [0, ∞] → [0, ∞] used to measure quantitatively the <a href="Uniform_continuity" title="Uniform continuity">uniform continuity</a> of functions. So, a function f : I → R admits ω as a modulus of continuity if

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<annotation encoding="application/x-tex">{\displaystyle |f(x)-f(y)|\leq \omega (|x-y|),}</annotation>

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</math><img src="./0352d5080a7cffe47e4cb400fccdbe395f480fd3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.415ex; height:2.843ex;" alt="{\displaystyle |f(x)-f(y)|\leq \omega (|x-y|),}" loading="lazy"></dd></dl>

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 <a href="Equicontinuity" title="Equicontinuity">equicontinuous families</a>. For instance, the modulus ω(t) := kt describes the k-<a href="Lipschitz_functions" class="mw-redirect" title="Lipschitz functions">Lipschitz functions</a>, the moduli ω(t) := ktα describe the <a href="H%C3%B6lder_continuity" class="mw-redirect" title="Hölder continuity">Hölder continuity</a>, 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 <a href="(%CE%B5%2C_%CE%B4)-definition_of_limit" class="mw-redirect" title="(ε, δ)-definition of limit">(ε, δ) definition of uniform continuity</a>. The same notions generalize naturally to functions between <a href="Metric_space" title="Metric space">metric spaces</a>. 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 <a href="Linear_growth" class="mw-redirect" title="Linear growth">growth</a>). 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

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<annotation encoding="application/x-tex">{\displaystyle {\frac {d_{Y}(f(x),f(x'))}{d_{X}(x,x')}}}</annotation>

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</math><img src="./7f1e347319057db1c70afcdd850184fd9c645070.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:15.894ex; height:6.509ex;" alt="{\displaystyle {\frac {d_{Y}(f(x),f(x'))}{d_{X}(x,x')}}}" loading="lazy"></dd></dl>

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.

<div class="mw-heading mw-heading2"><h2 id="Formal_definition">Formal definition</h2></div>

Formally, a modulus of continuity is any increasing real-extended valued function ω : [0, ∞] → [0, ∞], vanishing at 0 and continuous at 0, that is

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<annotation encoding="application/x-tex">{\displaystyle \lim _{t\to 0}\omega (t)=\omega (0)=0.}</annotation>

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</math><img src="./047144003ceff9e19477611e139c681a6155c1c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.135ex; height:4.009ex;" alt="{\displaystyle \lim _{t\to 0}\omega (t)=\omega (0)=0.}" loading="lazy"></dd></dl>

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,

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<annotation encoding="application/x-tex">{\displaystyle \forall x'\in X:d_{Y}(f(x),f(x'))\leq \omega (d_{X}(x,x')).}</annotation>

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</math><img src="./92ee594e345b5093e82dbfbcac3a03055073dc8d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.152ex; height:3.009ex;" alt="{\displaystyle \forall x'\in X:d_{Y}(f(x),f(x'))\leq \omega (d_{X}(x,x')).}" loading="lazy"></dd></dl>

Also, f admits ω as (global) modulus of continuity if and only if,

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<annotation encoding="application/x-tex">{\displaystyle \forall x,x'\in X:d_{Y}(f(x),f(x'))\leq \omega (d_{X}(x,x')).}</annotation>

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</math><img src="./a119c589293dc7511876993274f71164d51fdc72.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.516ex; height:3.009ex;" alt="{\displaystyle \forall x,x'\in X:d_{Y}(f(x),f(x'))\leq \omega (d_{X}(x,x')).}" loading="lazy"></dd></dl>

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.

<div class="mw-heading mw-heading3"><h3 id="Elementary_facts">Elementary facts</h3></div>

<ul><li>If f has ω as modulus of continuity and ω1 ≥ ω, then f admits ω1 too as modulus of continuity.</li>

<li>If f : X → Y and g : Y → Z are functions between metric spaces with moduli respectively ω1 and ω2 then the composition map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\circ f:X\to Z}">

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<annotation encoding="application/x-tex">{\displaystyle g\circ f:X\to Z}</annotation>

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</math><img src="./5e5cfd853da7199884ac0460cecd0b281f9846d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.801ex; height:2.509ex;" alt="{\displaystyle g\circ f:X\to Z}" loading="lazy"> has modulus of continuity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{2}\circ \omega _{1}}">

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<annotation encoding="application/x-tex">{\displaystyle \omega _{2}\circ \omega _{1}}</annotation>

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</math><img src="./da8e9d25c3797aca47868b1c8fdc85a9053b8ada.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.195ex; height:2.009ex;" alt="{\displaystyle \omega _{2}\circ \omega _{1}}" loading="lazy">.</li>

<li>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 <a href="Pointwise_convergence" title="Pointwise convergence">pointwise convergence</a>.</li>

<li>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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|g\|_{\infty }\omega _{1}+\|f\|_{\infty }\omega _{2}}">

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<annotation encoding="application/x-tex">{\displaystyle \|g\|_{\infty }\omega _{1}+\|f\|_{\infty }\omega _{2}}</annotation>

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</math><img src="./bab4c579360e5c87bceb6b34e26345002c0ca04e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.636ex; height:2.843ex;" alt="{\displaystyle \|g\|_{\infty }\omega _{1}+\|f\|_{\infty }\omega _{2}}" loading="lazy">.</li>

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<annotation encoding="application/x-tex">{\displaystyle \{f_{\lambda }\}_{\lambda \in \Lambda }}</annotation>

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</math><img src="./878a66df9b673f35b8396ae7d6784cd12ae98b4c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.082ex; height:2.843ex;" alt="{\displaystyle \{f_{\lambda }\}_{\lambda \in \Lambda }}" loading="lazy"> is a family of real-valued functions on the metric space X with common modulus of continuity ω, then the inferior envelope <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \inf _{\lambda \in \Lambda }f_{\lambda }}">

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<annotation encoding="application/x-tex">{\displaystyle \inf _{\lambda \in \Lambda }f_{\lambda }}</annotation>

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</math><img src="./a834924ed74c84a17a6fef51692cb454c157a73c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:5.912ex; height:4.009ex;" alt="{\displaystyle \inf _{\lambda \in \Lambda }f_{\lambda }}" loading="lazy">, respectively, the superior envelope <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sup _{\lambda \in \Lambda }f_{\lambda }}">

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<annotation encoding="application/x-tex">{\displaystyle \sup _{\lambda \in \Lambda }f_{\lambda }}</annotation>

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</math><img src="./b5f9dd441f82e9b2efb918578143fb3be8774270.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:6.218ex; height:4.343ex;" alt="{\displaystyle \sup _{\lambda \in \Lambda }f_{\lambda }}" loading="lazy">, 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.</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Remarks">Remarks</h3></div>

<ul><li>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, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{1}(t):=\sup _{s\leq t}\omega (s)}">

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<annotation encoding="application/x-tex">{\displaystyle \omega _{1}(t):=\sup _{s\leq t}\omega (s)}</annotation>

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</math> is increasing, and ω1 ≥ ω; <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{2}(t):={\frac {1}{t}}\int _{t}^{2t}\omega _{1}(s)ds}">

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<annotation encoding="application/x-tex">{\displaystyle \omega _{2}(t):={\frac {1}{t}}\int _{t}^{2t}\omega _{1}(s)ds}</annotation>

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</math> is also continuous, and ω2 ≥ ω1, and a suitable variant of the preceding definition also makes ω2 infinitely differentiable in [0, ∞].</li>

<li>Any uniformly continuous function admits a minimal modulus of continuity ωf, that is sometimes referred to as the (optimal) modulus of continuity of f: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{f}(t):=\sup\{d_{Y}(f(x),f(x')):x\in X,x'\in X,d_{X}(x,x')\leq t\},\quad \forall t\geq 0.}">

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<annotation encoding="application/x-tex">{\displaystyle \omega _{f}(t):=\sup\{d_{Y}(f(x),f(x')):x\in X,x'\in X,d_{X}(x,x')\leq t\},\quad \forall t\geq 0.}</annotation>

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</math> 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) : <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{f}(t;x):=\sup\{d_{Y}(f(x),f(x')):x'\in X,d_{X}(x,x')\leq t\},\quad \forall t\geq 0.}">

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<annotation encoding="application/x-tex">{\displaystyle \omega _{f}(t;x):=\sup\{d_{Y}(f(x),f(x')):x'\in X,d_{X}(x,x')\leq t\},\quad \forall t\geq 0.}</annotation>

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</math> 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.</li>

<li>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 → R such that f(n) := n2 is uniformly continuous with respect to the <a href="Discrete_metric" class="mw-redirect" title="Discrete metric">discrete metric</a> on N, and its minimal modulus of continuity is ωf(t) = +∞ for any t≥1, and ωf(t) = 0 otherwise. However, the situation is different for uniformly continuous functions defined on compact or convex subsets of normed spaces.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Special_moduli_of_continuity">Special moduli of continuity</h2></div>

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 <a href="Concave_function" title="Concave function">concave</a>, or <a href="Subadditive" class="mw-redirect" title="Subadditive">subadditive</a>, 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:

<ul><li>ω is concave;</li>

<li>ω is subadditive;</li>

<li>ω is uniformly continuous;</li>

<li>ω is sublinear, that is, there are constants a and b such that ω(t) ≤ at+b for all t;</li>

<li>ω is dominated by a concave modulus, that is, there exists a concave modulus of continuity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {\omega }}}">

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<annotation encoding="application/x-tex">{\displaystyle {\tilde {\omega }}}</annotation>

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</math><img src="./e60cbeebb31255ddce04c7526d2631bfd49fb8bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:2.176ex;" alt="{\displaystyle {\tilde {\omega }}}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega (t)\leq {\tilde {\omega }}(t)}">

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<annotation encoding="application/x-tex">{\displaystyle \omega (t)\leq {\tilde {\omega }}(t)}</annotation>

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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 <a href="Convex_set" title="Convex set">convex set</a> C of a normed space E always admits a <a href="Subadditive" class="mw-redirect" title="Subadditive">subadditive</a> 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:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\omega _{f}(s+t)&=\sup _{|x-x'|\leq t+s}d_{Y}(f(x),f(x'))\\&\leq \sup _{|x-x'|\leq t+s}\left\{d_{Y}\left(f(x),f\left(x-t{\frac {x-x'}{|x-x'|}}\right)\right)+d_{Y}\left(f\left(x-t{\frac {x-x'}{|x-x'|}}\right),f(x')\right)\right\}\\&\leq \omega _{f}(t)+\omega _{f}(s).\end{aligned}}}</annotation>

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</math><img src="./b2aadb632014b21b96fdf69b5d21b511694cc2a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.846ex; margin-bottom: -0.326ex; width:90.056ex; height:15.509ex;" alt="{\displaystyle {\begin{aligned}\omega _{f}(s+t)&=\sup _{|x-x'|\leq t+s}d_{Y}(f(x),f(x'))\\&\leq \sup _{|x-x'|\leq t+s}\left\{d_{Y}\left(f(x),f\left(x-t{\frac {x-x'}{|x-x'|}}\right)\right)+d_{Y}\left(f\left(x-t{\frac {x-x'}{|x-x'|}}\right),f(x')\right)\right\}\\&\leq \omega _{f}(t)+\omega _{f}(s).\end{aligned}}}" loading="lazy"></dd></dl>

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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{Y}(f(x),f(x'))/d_{X}(x,x')}">

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<annotation encoding="application/x-tex">{\displaystyle d_{Y}(f(x),f(x'))/d_{X}(x,x')}</annotation>

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</math><img src="./5752e7ec11d70ea5fd0a19650b9abcaadd1b3b76.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.249ex; height:3.009ex;" alt="{\displaystyle d_{Y}(f(x),f(x'))/d_{X}(x,x')}" loading="lazy"> 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.

<div class="mw-heading mw-heading3"><h3 id="Sublinear_moduli,_and_bounded_perturbations_from_Lipschitz">Sublinear moduli, and bounded perturbations from Lipschitz</h3></div>

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 modulus 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).

<div class="mw-heading mw-heading3"><h3 id="Subadditive_moduli,_and_extendibility">Subadditive moduli, and extendibility</h3></div>

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:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}f_{*}(x)&:=\sup _{y\in X}\left\{f(y)-\omega (|x-y|)\right\},\\f^{*}(x)&:=\inf _{y\in X}\left\{f(y)+\omega (|x-y|)\right\}.\end{aligned}}}</annotation>

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</math><img src="./97c649f1c696fe29505f505aff80141e390f762c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:34.216ex; height:9.509ex;" alt="{\displaystyle {\begin{aligned}f_{*}(x)&:=\sup _{y\in X}\left\{f(y)-\omega (|x-y|)\right\},\\f^{*}(x)&:=\inf _{y\in X}\left\{f(y)+\omega (|x-y|)\right\}.\end{aligned}}}" loading="lazy"></dd></dl>

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 <a href="Kuratowski_embedding" title="Kuratowski embedding">Kuratowski embedding</a> 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 <a href="Tietze_extension_theorem" title="Tietze extension theorem">Tietze extension theorem</a> 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 <a href="Kirszbraun_theorem" title="Kirszbraun theorem">Kirszbraun theorem</a>.

<div class="mw-heading mw-heading3"><h3 id="Concave_moduli_and_Lipschitz_approximation">Concave moduli and Lipschitz approximation</h3></div>

Every special uniformly continuous real-valued function f : X → R defined on the metric space X is <a href="Uniform_convergence" title="Uniform convergence">uniformly</a> 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

<dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega (t)=\inf {\big \{}at+b\,:\,a>0,\,b>0,\,\forall x\in X,\,\forall x'\in X\,\,|f(x)-f(x')|\leq ad(x,x')+b{\big \}}.}">

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<annotation encoding="application/x-tex">{\displaystyle \omega (t)=\inf {\big \{}at+b\,:\,a>0,\,b>0,\,\forall x\in X,\,\forall x'\in X\,\,|f(x)-f(x')|\leq ad(x,x')+b{\big \}}.}</annotation>

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</math><img src="./45f731e6f8f0d2b86ec6e238d443a625c8b8af2c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:82.788ex; height:3.176ex;" alt="{\displaystyle \omega (t)=\inf {\big \{}at+b\,:\,a>0,\,b>0,\,\forall x\in X,\,\forall x'\in X\,\,|f(x)-f(x')|\leq ad(x,x')+b{\big \}}.}" loading="lazy"></dd></dl>

Let δ(s) be the uniform <a href="Metric_spaces" class="mw-redirect" title="Metric spaces">distance</a> between the function f and the set Lips of all Lipschitz real-valued functions on C having Lipschitz constant s :

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<annotation encoding="application/x-tex">{\displaystyle \delta (s):=\inf {\big \{}\|f-u\|_{\infty ,X}\,:\,u\in \mathrm {Lip} _{s}{\big \}}\leq +\infty .}</annotation>

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</math><img src="./24c77a0309f4c2bed9a47b6bf3b94dd765fb45fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:44.103ex; height:3.176ex;" alt="{\displaystyle \delta (s):=\inf {\big \{}\|f-u\|_{\infty ,X}\,:\,u\in \mathrm {Lip} _{s}{\big \}}\leq +\infty .}" loading="lazy"></dd></dl>

Then the functions ω(t) and δ(s) can be related with each other via a <a href="Legendre_transformation" title="Legendre transformation">Legendre transformation</a>: more precisely, the functions 2δ(s) and −ω(−t) (suitably extended to +∞ outside their domains of finiteness) are a pair of conjugated convex functions,<a href="#cite_note-1">[1]</a> for

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<annotation encoding="application/x-tex">{\displaystyle 2\delta (s)=\sup _{t\geq 0}\left\{\omega (t)-st\right\},}</annotation>

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</math><img src="./b081c893f33737ba1e72360cd93a0d7cff6e7993.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:24.322ex; height:4.676ex;" alt="{\displaystyle 2\delta (s)=\sup _{t\geq 0}\left\{\omega (t)-st\right\},}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle \omega (t)=\inf _{s\geq 0}\left\{2\delta (s)+st\right\}.}</annotation>

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</math><img src="./d5aa1d962df5f6e89d692a15ca84dcbc8ffc45ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:23.692ex; height:4.176ex;" alt="{\displaystyle \omega (t)=\inf _{s\geq 0}\left\{2\delta (s)+st\right\}.}" loading="lazy"></dd></dl>

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

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<annotation encoding="application/x-tex">{\displaystyle f_{s}:=\delta (s)+\inf _{y\in X}\{f(y)+sd(x,y)\},\quad \mathrm {for} \ s\in \mathrm {dom} (\delta ):}</annotation>

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</math><img src="./35b8beaea651e100c2a36ef8502275d65f8c1b26.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:52.382ex; height:4.343ex;" alt="{\displaystyle f_{s}:=\delta (s)+\inf _{y\in X}\{f(y)+sd(x,y)\},\quad \mathrm {for} \ s\in \mathrm {dom} (\delta ):}" loading="lazy"></dd></dl>

each function fs has Lipschitz constant s and

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<annotation encoding="application/x-tex">{\displaystyle \|f-f_{s}\|_{\infty ,X}=\delta (s);}</annotation>

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</math><img src="./a74578d0210f08313596750ac78daf6279ce53c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.013ex; height:3.009ex;" alt="{\displaystyle \|f-f_{s}\|_{\infty ,X}=\delta (s);}" loading="lazy"></dd></dl>

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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(s^{-{\frac {\alpha }{1-\alpha }}}),}">

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<annotation encoding="application/x-tex">{\displaystyle O(s^{-{\frac {\alpha }{1-\alpha }}}),}</annotation>

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</math><img src="./2a68c026d23201e5145c82b59f87b68e41be5c97.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.226ex; height:3.843ex;" alt="{\displaystyle O(s^{-{\frac {\alpha }{1-\alpha }}}),}" loading="lazy"> while the almost Lipschitz functions are characterized by an exponential speed of convergence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(e^{-as}).}">

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<annotation encoding="application/x-tex">{\displaystyle O(e^{-as}).}</annotation>

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</math><img src="./3da1c216dee4d1dd826ea06ffbd4276d5f34cf97.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.464ex; height:3.009ex;" alt="{\displaystyle O(e^{-as}).}" loading="lazy">

<div class="mw-heading mw-heading2"><h2 id="Examples_of_use">Examples of use</h2></div>

<ul><li>Let f : [a, b] → R a continuous function. In the proof that f is <a href="Riemann_integrable" class="mw-redirect" title="Riemann integrable">Riemann integrable</a>, one usually bounds the distance between the upper and lower <a href="Riemann_sums" class="mw-redirect" title="Riemann sums">Riemann sums</a> with respect to the Riemann partition P := {t0, ..., tn} in terms of the modulus of continuity of f and the <a href="Riemann_integrable" class="mw-redirect" title="Riemann integrable">mesh</a> of the partition P (which is the number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle |P|:=\max _{0\leq i<n}(t_{i+1}-t_{i})}">

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<annotation encoding="application/x-tex">{\textstyle |P|:=\max _{0\leq i<n}(t_{i+1}-t_{i})}</annotation>

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</math><img src="./952a517e610cf6989d4d47c9fd190691fa62b234.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.304ex; height:2.843ex;" alt="{\textstyle |P|:=\max _{0\leq i<n}(t_{i+1}-t_{i})}" loading="lazy">) <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{*}(f;P)-S_{*}(f;P)\leq (b-a)\omega (|P|).}">

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<annotation encoding="application/x-tex">{\displaystyle S^{*}(f;P)-S_{*}(f;P)\leq (b-a)\omega (|P|).}</annotation>

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<li>For an example of use in the Fourier series, see <a href="Dini_test" title="Dini test">Dini test</a>.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

Steffens (2006, p. 160) attributes the first usage of omega for the modulus of continuity to <a href="Lebesgue" class="mw-redirect" title="Lebesgue">Lebesgue</a> (1909, p. 309/p. 75) where omega refers to the oscillation of a Fourier transform. <a href="De_la_Vall%C3%A9e_Poussin" class="mw-redirect" title="De la Vallée Poussin">De la Vallée Poussin</a> (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".

<div class="mw-heading mw-heading2"><h2 id="The_translation_group_of_Lp_functions,_and_moduli_of_continuity_Lp.">The translation group of Lp functions, and moduli of continuity Lp.</h2></div>

Let 1 ≤ p; let f : Rn → R a function of class Lp, and let h ∈ Rn. The h-<a href="Translation_(geometry)" title="Translation (geometry)">translation</a> 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:

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<annotation encoding="application/x-tex">{\displaystyle \|\tau _{h}f-f\|_{p}=o(1).}</annotation>

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</math><img src="./3ad9bdd6c7a176ea2767d6ea1c1828eb6dc012c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.822ex; height:3.009ex;" alt="{\displaystyle \|\tau _{h}f-f\|_{p}=o(1).}" loading="lazy"></dd></dl>

Therefore, since translations are in fact linear isometries, also

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<annotation encoding="application/x-tex">{\displaystyle \|\tau _{v+h}f-\tau _{v}f\|_{p}=o(1),}</annotation>

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</math><img src="./df7c1afa44a83613d93174d260adb1440daf4b35.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.943ex; height:3.009ex;" alt="{\displaystyle \|\tau _{v+h}f-\tau _{v}f\|_{p}=o(1),}" loading="lazy"></dd></dl>

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

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<annotation encoding="application/x-tex">{\displaystyle \|\tau _{h}f-f\|_{p}\leq \omega (h).}</annotation>

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</math><img src="./7c5d4447980e460a8443360ed76fc9ac1362d59e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.316ex; height:3.009ex;" alt="{\displaystyle \|\tau _{h}f-f\|_{p}\leq \omega (h).}" loading="lazy"></dd></dl>

This way, moduli of continuity also give a quantitative account of the continuity property shared by all Lp functions.

<div class="mw-heading mw-heading2"><h2 id="Modulus_of_continuity_of_higher_orders">Modulus of continuity of higher orders</h2></div>

It can be seen that formal definition of the modulus uses notion of <a href="Finite_difference" title="Finite difference">finite difference</a> of first order:

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<annotation encoding="application/x-tex">{\displaystyle \omega _{f}(\delta )=\omega (f,\delta )=\sup \limits _{x;|h|<\delta ;}\left|\Delta _{h}(f,x)\right|.}</annotation>

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</math><img src="./56a01d497f5e8f80a935c710e5d3c68e79035fdf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:35.271ex; height:5.009ex;" alt="{\displaystyle \omega _{f}(\delta )=\omega (f,\delta )=\sup \limits _{x;|h|<\delta ;}\left|\Delta _{h}(f,x)\right|.}" loading="lazy"></dd></dl>

If we replace that difference with a <a href="Finite_difference#Higher-order_differences" title="Finite difference">difference of order n</a>, we get a modulus of continuity of order n:

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<annotation encoding="application/x-tex">{\displaystyle \omega _{n}(f,\delta )=\sup \limits _{x;|h|<\delta ;}\left|\Delta _{h}^{n}(f,x)\right|.}</annotation>

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</math><img src="./9f5da2b10fad494d56f189b203f3c7217d758c66.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:27.99ex; height:5.176ex;" alt="{\displaystyle \omega _{n}(f,\delta )=\sup \limits _{x;|h|<\delta ;}\left|\Delta _{h}^{n}(f,x)\right|.}" loading="lazy"></dd></dl>

<ul><li><a href="Constructive_analysis" title="Constructive analysis">Constructive analysis</a></li>

<li><a href="Modulus_of_convergence" title="Modulus of convergence">Modulus of convergence</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-1"><a href="#cite_ref-1">^</a> <a rel="nofollow" class="external text" href="https://mathoverflow.net/q/194890">Legendre transform and Lipschitz approximation</a>

</li>

</ol></div></div>

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</style><cite id="CITEREFChoquet1969" class="citation book cs1 cs1-prop-foreign-lang-source">Choquet, G. (1969). Topologie : espaces topologiques et espaces métriques, fonctions numériques, espaces vectoriels topologiques. Cours D'Analyse (in French). Vol. 2 (2nd ed.). Paris: Masson et Cie. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/299915897">299915897</a>.</cite></li>

<li><cite id="CITEREFEfimov2001" class="citation cs2">Efimov, A. V. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Continuity,_modulus_of">"Continuity, modulus of"</a>, <a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, Springer, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>1-4020-0609-8</bdi></cite></li>

<li><cite id="CITEREFLebesgue1909" class="citation journal cs1">Lebesgue, H. (1909). "Sur les intégrales singulières". Annales de la Faculté des Sciences de Toulouse: Mathématiques. 3 (1): 25–117. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.5802%2Fafst.257">10.5802/afst.257</a>.</cite> Reproduced in: <cite id="CITEREFLebesgue" class="citation book cs1 cs1-prop-foreign-lang-source">Lebesgue, Henri. Œuvres scientifiques (in French). Vol. 3. pp. 259–351.</cite></li>

<li><cite id="CITEREFPoussin1952" class="citation book cs1 cs1-prop-foreign-lang-source">Poussin, Ch. de la Vallée (1952) [1919]. L'approximation des fonctions d'une variable réelle (in French). Paris: Gauthier-Villars.</cite></li>

<li><cite id="CITEREFBenyaminiLindenstrauss1998" class="citation book cs1">Benyamini, Y; Lindenstrauss, J (1998). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=lXZ95EKwjYUC&pg=PP13">Geometric Nonlinear Functional Analysis: Volume 1</a>. Colloquium Publications. Vol. 48. American Mathematical Society. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8218-6963-5</bdi>.</cite></li>

<li><cite id="CITEREFSteffens2006" class="citation book cs1">Steffens, K.-G. (2006). The History of Approximation Theory. Boston: Birkhäuser. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-8176-4353-2</bdi>.</cite></li></ul></div><div><div class="zim-footer">

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