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One of the basic characteristics of continuous functions. The modulus of continuity of a continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025580/c0255801.png" /> on a closed interval is defined, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025580/c0255802.png" />, as
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{{MSC|54C05}}
 +
[[Category:Analysis]]
 +
{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025580/c0255803.png" /></td> </tr></table>
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One of the basic characteristics of (uniformly) continuous functions. The modulus of a continuity of a continuous function $f:\mathbb R^n \supset E \to \mathbb R^k$ is given by
 +
\[
 +
\omega (\delta, f) := \sup_{|x-y|\leq \delta} |f(x)-f(y)|\, ,
 +
\]
 +
where we are implicitely assuming that $\lim_{\delta\downarrow 0} \omega (\delta, f) = 0$, which is indeed true if and ony if $f$ is the [[Uniform continuity|uniformly continuous]]. The definition of the modulus of continuity was introduced by H. Lebesgue in 1910 for functions of one real variable, although in essence the concept was known earlier. The definition can be readily extended to uniformly continuous maps $f$ between [[Metric space|metric spaces]] $(X,d)$ and $(Y, \delta)$, simply setting
 +
\[
 +
\omega (\delta, f) := \sup_{d(x,y)\leq \delta} \delta (f(x), f(y))\, .
 +
\]
  
The definition of the modulus of continuity was introduced by H. Lebesgue in 1910, although in essence the concept was known earlier. If the modulus of continuity of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025580/c0255804.png" /> satisfies the condition
+
====Examples====
 +
Very special classes of moduli of continuity give notable classes of functions. For instance
 +
* If $\omega (\delta, f) \leq M \delta$ for some constant $M>0$, then $f$ satisfies the [[Lipschitz condition]]; the least constant $M$ for which such inequality holds is the [[Lipschitz constant]] of $f$; cf. also [[Lipschitz function]].
 +
* If $\omega (\delta, f) \leq M \delta^\alpha$ for some constants $M>0$, $\alpha\in ]0,1]$, then $f$ satisfies the [[Hölder condition]] of exponent $\alpha$.
 +
* If
 +
\[
 +
\int_0^1 \frac{\omega (\delta, f)}{\delta}\, d\delta < \infty\, ,
 +
\]
 +
then $f$ is called, by some authors, ''Dini continuous'' (such condition plays a special role in the convergence of Fourier series, cf. [[Dini criterion]]).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025580/c0255805.png" /></td> </tr></table>
+
====Basic properties====
 +
It is elementary to derive bounds on the modulus of continuity of linear combinations, compositions and infima of uniformly continuous functions in term of their respective moduli of continuity. In particular
 +
* $\omega (\delta, \lambda f + \mu g) \leq |\lambda| \omega (\delta, f) + |\mu| \omega (\delta, g)$;
 +
* $\omega (\delta, g\circ f) \leq \omega (\omega (\delta, f), g)$;
 +
* If $\{f_\lambda\}$ is a family of real-valued functions with $\omega (\delta, f_\lambda) \leq h (\delta)$ for some common function $h$, then $\inf_\lambda f_\lambda$ and $\sup_\lambda f_\lambda$ are also continuous with modulus of continuity bounded by $h$, provided the respective infima and suprema are finite (for which it is indeed necessary and sufficient that they are finite at some point).
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025580/c0255806.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025580/c0255807.png" /> is said to satisfy a [[Lipschitz condition|Lipschitz condition]] of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025580/c0255808.png" />.
+
For a non-negative function $\omega: [0,\infty[\to \mathbb R$ there is a continuous function $f: [0, \infty[ \to \mathbb R$ with $\omega (\delta, f) = \omega (\delta)$ for every $\delta$ if and only if the following properties hold:
 +
* $\omega (0) = 0$
 +
* $\omega$ is continuous
 +
* $\omega$ is subadditive, namely $\omega (\alpha + \beta) \leq \omega (\alpha) + \omega (\beta)$ for every $\alpha, \beta \geq 0$.
  
For a non-negative function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025580/c0255809.png" /> defined for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025580/c02558010.png" /> to be the modulus of continuity of some continuous function it is necessary and sufficient that it has the following properties: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025580/c02558011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025580/c02558012.png" /> is non-decreasing, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025580/c02558013.png" /> is continuous, and
+
====Generalizations====
 +
As already mentioned, the notion can be easily generalized to maps between arbitrary metric spaces.  
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025580/c02558014.png" /></td> </tr></table>
+
One can also consider moduli of continuity of higher orders. For instance, if $f$ is a function of one variable, the modulus of continuity can be rewritten as
 +
\[
 +
\omega (\delta, f) = \sup_{|h| \leq \delta} \max_x \Delta_h f (x)\, ,
 +
\]
 +
where $\Delta_h f (x) = |f(x+h)-f(x)|$ is the usual [[Finite-difference calculus|finite difference]] of first order. Therefore, if we introduce the higher oder finite differences
 +
\[
 +
\Delta^k_h (x) = \sum_{i =0}^k (-1)^{k-i} {{k}\choose{i}} f (x+ih)\, ,
 +
\]
 +
the higher oder moduli of continuity can be defined as
 +
\[
 +
\omega_k (\delta, f) = \sup_{|h|\leq \delta} \max_x \Delta^k_h f (x)\, .
 +
\]
 +
See also [[Smoothness, modulus of]]. Moduli of continuity and smoothness are extensively used in [[Approximation theory|approximation theory]] and Fourier analysis (cf. [[Harmonic analysis|Harmonic analysis]]).
  
One can also consider moduli of continuity of higher orders,
+
A further common generalization replaces pointwise maxima with integrals. For instance, if $f: \mathbb R^n \to \mathbb R$ is Lebesgue measurable, the $L^p$-modulus of continuity of $f$ is defined as
 +
\[
 +
\omega^{(p)} (\delta, f) := \sup_{|\xi|\leq \delta} \int |f(x+\xi) - f(x)|^p\, dx\, .
 +
\]
 +
An obvious variant can be defined for maps on open domains $\Omega$ by simply restricting the domain of integration to $\{x\in \Omega : {\rm dist}\, (x, \partial \Omega) < \delta\}$. A classical characterization of the [[Sobolev space]] $W^{1,p} (\mathbb R^n)$ is then the following
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025580/c02558015.png" /></td> </tr></table>
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'''Theorem'''
 +
Let $p \in ]1, \infty[$. $f\in L^p (\mathbb R^n)$ belongs to $W^{1,p}$ if and only if there is a constant $M$ such that $\omega^{(p)} (\delta, f) \leq M \delta^p$ for every $\delta$.
  
where
+
Cf. Theorem 3 in Section 5.8 of {{Cite|Ev}}. The limiting case $p=\infty$ of the above theorem is also valid and gives then the identity between $W^{1,\infty} (\mathbb R^n)$ and the space ${\rm Lip}_b (\mathbb R^n)$ of bounded Lipschitz functions. For $p=1$ the property $\omega^{(1)} (\delta, f) \leq M \delta$ characterizes instead the space of [[Function of bounded variation|functions of bounded variation]].
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025580/c02558016.png" /></td> </tr></table>
 
 
 
is the finite difference of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025580/c02558018.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025580/c02558019.png" />, and moduli of continuity in arbitrary function spaces, for example, the integral modulus of continuity of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025580/c02558020.png" /> that is integrable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025580/c02558021.png" /> to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025580/c02558022.png" />-th power, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025580/c02558023.png" />:
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025580/c02558024.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
 
 
 
For a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025580/c02558025.png" />-periodic function the integral in (*) is taken over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025580/c02558026.png" />.
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Zygmund,  "Trigonometric series" , '''1''' , Cambridge Univ. Press  (1988)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.I. [N.I. Akhiezer] Achiezer,  "Theory of approximation" , F. Ungar  (1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"V.K. Dzyadyk,  "Introduction to the theory of uniform approximation of functions by polynomials" , Moscow  (1977)  (In Russian)</TD></TR></table>
+
{|
 
+
|-
 
+
|valign="top"|{{Ref|AGS}}|| N.I. [N.I. Akhiezer] Achiezer,  "Theory of approximation" , F. Ungar  (1956)  (Translated from Russian)
 +
|-
 +
|valign="top"|{{Ref|AFP}}||    L. Ambrosio, N.  Fusco, D.  Pallara, "Functions of bounded variations  and  free  discontinuity  problems". Oxford Mathematical Monographs. The    Clarendon Press,  Oxford University Press, New York, 2000.      {{MR|1857292}}{{ZBL|0957.49001}}
 +
|-
 +
|valign="top"|{{Ref|Dz}}|| V.K. Dzyadyk,  "Introduction to the theory of uniform approximation of functions by polynomials" , Moscow  (1977)  (In Russian)  
  
====Comments====
+
|-
See also [[Smoothness, modulus of|Smoothness, modulus of]]. Moduli of continuity and smoothness are extensively used in [[Approximation theory|approximation theory]] and Fourier analysis (cf. [[Harmonic analysis|Harmonic analysis]]).
+
|valign="top"|{{Ref|Ev}}||  L.C. Evans, "Partial differential equations", Graduate studies in mathematics. American Mathematical Society (1998).
 +
|-
 +
 
 +
|valign="top"|{{Ref|St}}|| K. G. Steffens, "The History of Approximation Theory", Birkhäuser (2006).
 +
|-
 +
|valign="top"|{{Ref|Zy}}|| A. Zygmund,  "Trigonometric series" , '''1''' , Cambridge Univ. Press  (1988).
 +
|-
 +
|}

Revision as of 13:56, 11 November 2013

2020 Mathematics Subject Classification: Primary: 54C05 [MSN][ZBL]

One of the basic characteristics of (uniformly) continuous functions. The modulus of a continuity of a continuous function $f:\mathbb R^n \supset E \to \mathbb R^k$ is given by \[ \omega (\delta, f) := \sup_{|x-y|\leq \delta} |f(x)-f(y)|\, , \] where we are implicitely assuming that $\lim_{\delta\downarrow 0} \omega (\delta, f) = 0$, which is indeed true if and ony if $f$ is the uniformly continuous. The definition of the modulus of continuity was introduced by H. Lebesgue in 1910 for functions of one real variable, although in essence the concept was known earlier. The definition can be readily extended to uniformly continuous maps $f$ between metric spaces $(X,d)$ and $(Y, \delta)$, simply setting \[ \omega (\delta, f) := \sup_{d(x,y)\leq \delta} \delta (f(x), f(y))\, . \]

Examples

Very special classes of moduli of continuity give notable classes of functions. For instance

  • If $\omega (\delta, f) \leq M \delta$ for some constant $M>0$, then $f$ satisfies the Lipschitz condition; the least constant $M$ for which such inequality holds is the Lipschitz constant of $f$; cf. also Lipschitz function.
  • If $\omega (\delta, f) \leq M \delta^\alpha$ for some constants $M>0$, $\alpha\in ]0,1]$, then $f$ satisfies the Hölder condition of exponent $\alpha$.
  • If

\[ \int_0^1 \frac{\omega (\delta, f)}{\delta}\, d\delta < \infty\, , \] then $f$ is called, by some authors, Dini continuous (such condition plays a special role in the convergence of Fourier series, cf. Dini criterion).

Basic properties

It is elementary to derive bounds on the modulus of continuity of linear combinations, compositions and infima of uniformly continuous functions in term of their respective moduli of continuity. In particular

  • $\omega (\delta, \lambda f + \mu g) \leq |\lambda| \omega (\delta, f) + |\mu| \omega (\delta, g)$;
  • $\omega (\delta, g\circ f) \leq \omega (\omega (\delta, f), g)$;
  • If $\{f_\lambda\}$ is a family of real-valued functions with $\omega (\delta, f_\lambda) \leq h (\delta)$ for some common function $h$, then $\inf_\lambda f_\lambda$ and $\sup_\lambda f_\lambda$ are also continuous with modulus of continuity bounded by $h$, provided the respective infima and suprema are finite (for which it is indeed necessary and sufficient that they are finite at some point).

For a non-negative function $\omega: [0,\infty[\to \mathbb R$ there is a continuous function $f: [0, \infty[ \to \mathbb R$ with $\omega (\delta, f) = \omega (\delta)$ for every $\delta$ if and only if the following properties hold:

  • $\omega (0) = 0$
  • $\omega$ is continuous
  • $\omega$ is subadditive, namely $\omega (\alpha + \beta) \leq \omega (\alpha) + \omega (\beta)$ for every $\alpha, \beta \geq 0$.

Generalizations

As already mentioned, the notion can be easily generalized to maps between arbitrary metric spaces.

One can also consider moduli of continuity of higher orders. For instance, if $f$ is a function of one variable, the modulus of continuity can be rewritten as \[ \omega (\delta, f) = \sup_{|h| \leq \delta} \max_x \Delta_h f (x)\, , \] where $\Delta_h f (x) = |f(x+h)-f(x)|$ is the usual finite difference of first order. Therefore, if we introduce the higher oder finite differences \[ \Delta^k_h (x) = \sum_{i =0}^k (-1)^{k-i} {{k}\choose{i}} f (x+ih)\, , \] the higher oder moduli of continuity can be defined as \[ \omega_k (\delta, f) = \sup_{|h|\leq \delta} \max_x \Delta^k_h f (x)\, . \] See also Smoothness, modulus of. Moduli of continuity and smoothness are extensively used in approximation theory and Fourier analysis (cf. Harmonic analysis).

A further common generalization replaces pointwise maxima with integrals. For instance, if $f: \mathbb R^n \to \mathbb R$ is Lebesgue measurable, the $L^p$-modulus of continuity of $f$ is defined as \[ \omega^{(p)} (\delta, f) := \sup_{|\xi|\leq \delta} \int |f(x+\xi) - f(x)|^p\, dx\, . \] An obvious variant can be defined for maps on open domains $\Omega$ by simply restricting the domain of integration to $\{x\in \Omega : {\rm dist}\, (x, \partial \Omega) < \delta\}$. A classical characterization of the Sobolev space $W^{1,p} (\mathbb R^n)$ is then the following

Theorem Let $p \in ]1, \infty[$. $f\in L^p (\mathbb R^n)$ belongs to $W^{1,p}$ if and only if there is a constant $M$ such that $\omega^{(p)} (\delta, f) \leq M \delta^p$ for every $\delta$.

Cf. Theorem 3 in Section 5.8 of [Ev]. The limiting case $p=\infty$ of the above theorem is also valid and gives then the identity between $W^{1,\infty} (\mathbb R^n)$ and the space ${\rm Lip}_b (\mathbb R^n)$ of bounded Lipschitz functions. For $p=1$ the property $\omega^{(1)} (\delta, f) \leq M \delta$ characterizes instead the space of functions of bounded variation.

References

[AGS] N.I. [N.I. Akhiezer] Achiezer, "Theory of approximation" , F. Ungar (1956) (Translated from Russian)
[AFP] L. Ambrosio, N. Fusco, D. Pallara, "Functions of bounded variations and free discontinuity problems". Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000. MR1857292Zbl 0957.49001
[Dz] V.K. Dzyadyk, "Introduction to the theory of uniform approximation of functions by polynomials" , Moscow (1977) (In Russian)
[Ev] L.C. Evans, "Partial differential equations", Graduate studies in mathematics. American Mathematical Society (1998).
[St] K. G. Steffens, "The History of Approximation Theory", Birkhäuser (2006).
[Zy] A. Zygmund, "Trigonometric series" , 1 , Cambridge Univ. Press (1988).
How to Cite This Entry:
Continuity, modulus of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Continuity,_modulus_of&oldid=30701
This article was adapted from an original article by A.V. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article