Continuity, modulus of

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,

where

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 .

References

[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)

Comments

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