* Lévy's continuity theorem

** Lévy's continuity theorem on random variables

Such generalizations are useful for constructing different modulus of continuity.

Intuitively, a Lipschitz continuous function is limited in how fast it can change: for every pair of points on the graph of this function, the absolute value of the slope of the line connecting them is no greater than a definite real number ; this bound is called the function's " Lipschitz constant " ( or " modulus of uniform continuity ").

For example according to Errett Bishop's definitions, the continuity of a function ( such as sin x ) should be proved as a constructive bound on the modulus of continuity, meaning that the existential content of the assertion of continuity is a promise that can always be kept.

If and has modulus of continuity,

We define the local modulus of continuity at the point t by

The global modulus of continuity ( or simply the modulus of continuity ) is defined by

In mathematical analysis, a modulus of continuity is a function

So, a function admits as a modulus of continuity if and only if

Since moduli of continuity are required to be infinitesimal at 0, a function turns out to be uniformly continuous if and only if it admits a modulus of continuity.

Moreover, relevance to the notion is given by the fact that sets of functions sharing the same modulus of continuity are exactly equicontinuous families.

For instance, the modulus describes the k-Lipschitz functions, the moduli describe the Hölder continuity, the modulus describes the almost Lipschitz class, and so on.

For a function between metric spaces, it is equivalent to admit a modulus of continuity that is either concave, or subadditive, or uniformly continuous, or sublinear ( in the sense of growth ).

However, a uniformly continuous function on a general metric space admits a concave modulus of continuity if and only if the ratios are uniformly bounded for all pairs bounded away from the diagonal of.

Formally, a modulus of continuity is any real-extended valued function

A function admits as ( local ) modulus of continuity at the point if and only if,

Also, admits as ( global ) modulus of continuity if and only if,

One equivalently says that is a modulus of continuity ( resp., at ) for, or shortly, is-continuous ( resp., at ).