View source for Continuity, modulus of

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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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

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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 <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

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One can also consider moduli of continuity of higher orders,

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where

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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====
<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>



====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]]).