# Lipschitz continuity

In mathematics, a function

*f*:*I*→**R**

defined on an interval of real numbers with real values is called **Lipschitz continuous** (or is said to satisfy a **Lipschitz condition**) if there exists a constant

*K*≥ 0

such that

for all in the interval *I*. The smallest such *K* is called the **Lipschitz constant** of the map. The name is after the German mathematician Rudolf Lipschitz.

Intuitively, a Lipschitz continuous function is limited in how fast it can change; a line joining any two points on the graph of this function will never have a slope steeper than its Lipschitz constant *K*. The mean value theorem can be used to prove that any differentiable function with bounded derivative is Lipschitz continuous, with the Lipschitz constant being the largest magnitude of the derivative.

## Contents

## Examples

- The function defined on is Lipschitz continuous, with
*K*=14. This follows from the observation above.

- The function defined for all real numbers is Lipschitz continuous with the Lipschitz constant
*K*=1.

- The function defined on is Lipschitz continuous with the Lipschitz constant equal to 2. This is an example of a Lipschitz continuous function which is not differentiable.

- The function (the same function as in the first example) defined for all real numbers is
*not*Lipschitz continuous. This function becomes arbitrarily steep as*x*→∞.

- The function defined on is
*not*Lipschitz continuous. This function becomes infinitely steep as*x*→0 since its derivative becomes infinite.

## Lipschitz continuity in metric spaces

The notion of Lipschitz continuity can be extended to arbitrary metric spaces, when the absolute values in the definition is replaced by general distances. A function

*f*:*M*→*N*

between metric spaces *M* and *N* is called **Lipschitz continuous** if there exists a constant

*K*≥ 0

such that

- d(
*f*(*x*),*f*(*y*)) ≤*K*d(*x*,*y*)

for all *x* and *y* in *M*, with the smallest such *K* again being called the **Lipschitz constant**. Here, *d* denotes the distance function in the spaces *M* and *N*. The two distance functions could be different; the same notation was used because it is clear from the formula which distance is in which space.

If *f* : *M* → *N* satisfies the condition

- d(
*x*,*y*)/K ≤ d(*f*(*x*),*f*(*y*)) ≤*K*d(*x*,*y*)

where *K* ≥ 1 then *f* is called a **bilipschitz** function. Every bilipschitz function is injective. A bilipschitz bijection is the same thing as a Lipschitz bijection whose inverse function is also Lipschitz.

## Properties of Lipschitz continuous functions

Every Lipschitz continuous map is uniformly continuous, and hence *a fortiori* continuous.

Lipschitz continuous maps with Lipschitz constant *K* = 1 are called **short maps** and with
*K* < 1 are called *contraction mappings* if *M*=*N*; the latter are the subject of the Banach fixed point theorem.

Lipschitz continuity is an important condition in the existence and uniqueness theorem for ordinary differential equations.

If *U* is a subset of the metric space *M* and *f* : *U* → **R** is a Lipschitz continuous map, there always exist Lipschitz continuous maps *M* → **R** which extend *f* and have the same Lipschitz constant as *f* (see also Kirszbraun theorem).

A Lipschitz continuous map *f* : *I* → **R**, where *I* is an interval in **R**, is almost everywhere differentiable (everywhere except on a set of Lebesgue measure 0). If *K* is the Lipschitz constant of *f*, then |*f'*(*x*)| ≤ *K* whenever the derivative exists. Conversely, if *f* : *I* → **R** is a differentiable map with bounded derivative, |*f'*(*x*)| ≤ *L* for all *x* in *I*, then *f* is Lipschitz continuous with Lipschitz constant *K* ≤ *L*, a consequence of the mean value theorem.

All Banach spaces have the notion of Lipschitz continuity.

## Hölder continuity

If a map *f*: *M* → *N* satisfies the Lipschitz-like condition

*d*(*f*(*x*),*f*(*y*)) ≤*Kd*(*x*,*y*)^{α}

for some α > 0 (the *order*) and all *x*, *y*, it is said to be **Hölder-continuous** or *α-Hölder*.

## See also

de:Lipschitz-Stetigkeit es:Lipschitz continua fr:Application lipschitzienne he:תנאי ליפשיץ pl:Warunek Lipschitza sv:Lipschitzkontinuitet