Continuity, modulus of

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One of the basic characteristics of continuous functions. The modulus of continuity of a continuous function on a closed interval is defined, with , as

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 satisfies the condition

where , then is said to satisfy a Lipschitz condition of order .

For a non-negative function defined for to be the modulus of continuity of some continuous function it is necessary and sufficient that it has the following properties: , is non-decreasing, is continuous, and

One can also consider moduli of continuity of higher orders,


is the finite difference of order of , and moduli of continuity in arbitrary function spaces, for example, the integral modulus of continuity of a function that is integrable on to the -th power, :


For a -periodic function the integral in (*) is taken over .


[1] A. Zygmund, "Trigonometric series" , 1 , Cambridge Univ. Press (1988)
[2] N.I. [N.I. Akhiezer] Achiezer, "Theory of approximation" , F. Ungar (1956) (Translated from Russian)
[3] V.K. Dzyadyk, "Introduction to the theory of uniform approximation of functions by polynomials" , Moscow (1977) (In Russian)


See also Smoothness, modulus of. Moduli of continuity and smoothness are extensively used in approximation theory and Fourier analysis (cf. Harmonic analysis).

How to Cite This Entry:
Continuity, modulus of. Encyclopedia of Mathematics. URL:,_modulus_of&oldid=16325
This article was adapted from an original article by A.V. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article