# Continuity, modulus of

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

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 .

How to Cite This Entry:
Continuity, modulus of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Continuity,_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