Lipschitz continuity

Lipschitz continuity (after Rudolf Lipschitz ), and strain limitations, referred to in the Analysis a tightening of continuity. Intuitively, a Lipschitz continuous function only limited change quickly: All secants of a function having a slope whose absolute value is not larger than a constant, the Lipschitz constant.

A generalization of the Lipschitz continuity is the Hölder continuity.

Definition

A function is called Lipschitz continuous if there exists a constant such that

Applies to all.

This is a special case of the following general definition.

Be and metric spaces. A function is called Lipschitz continuous if there is a real number, so

Is satisfied. Lipschitz constant is called and it is always. Figuratively speaking, the magnitude of the slope of upward is limited by. If a function is Lipschitz continuous, then one can also say they satisfy the Lipschitz condition.

A weakening the Lipschitz continuity is the local Lipschitz continuity. A function is called locally Lipschitz continuous, if there is at any point in an environment, so that the restriction of this environment is Lipschitz continuous. A function that is defined only on a subset of words, Lipschitz or locally Lipschitz if it is Lipschitz or locally Lipschitz continuous with respect to the metric spaces and.

Properties

Lipschitz continuous functions are locally Lipschitz continuous ( choose all as environment and always as a Lipschitz constant ). Local Lipschitz continuous functions are continuous ( selected in the - definition of continuity ), and are in accordance with Lipschitz continuous functions uniformly continuous. Therefore, Lipschitz continuity " stronger" than uniform continuity. The converse is not true in general, so for example the function is indeed Hölder continuous with exponent and therefore uniformly continuous, but not Lipschitz continuous ( see example).

By the theorem of Rademacher a Lipschitz continuous function is differentiable almost everywhere. However, there are features which, although differentiated but not Lipschitz, eg. A differentiable function with Lipschitz continuous if and only if its first derivative is bounded.

Application

Lipschitz continuity is an important concept in the theory of ordinary differential equations to prove existence and uniqueness of solutions ( see Theorem of Picard - Lindelöf ). Pictures with a Lipschitz constant less than one is called contraction. These are important for the fixed point theorem of Banach.

Examples

For a Lipschitz continuous function is the quotient

With restricted by any Lipschitz constant of upwards. For locally Lipschitz continuous functions of the quotient is limited to sufficiently small neighborhoods.

Therefore, the function for

Although steady and even uniformly continuous but not locally Lipschitz continuous and hence not Lipschitz continuous.

For the function follows with

That

That is, is a Lipschitz constant for this function on the interval.

Because is the same for the quotient, it follows that only for a limited domain is Lipschitz continuous, but not for unlimited. The also by defined function is therefore not Lipschitz continuous.

The absolute value function, defined as

Is due to the reverse triangle inequality Lipschitz continuous with, but it is not ( at the site ) differentiable.