Lipschitz continuity

f : D → R

defined on a set D 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

|f(x)-f(y)|\le K |x-y|

for all $x, y$ in D. The smallest such K is called the Lipschitz constant of the function in that interval. 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, defined on an interval, that has a bounded derivative is Lipschitz continuous, with the Lipschitz constant being the supremum of the magnitude of the derivative.

Examples

The function $f(x)=x^2$ defined on $7$ is Lipschitz continuous, with K=14. This follows from the observation above.

The function $f(x)=\sqrt{x^2+5}$ defined for all real numbers is Lipschitz continuous with the Lipschitz constant K=1.

The function $f(x)=2|x-3|$ defined on $10$ is Lipschitz continuous with the Lipschitz constant equal to 2. This is an example of a Lipschitz continuous function that is not differentiable.

The function $f(x)=x^2$ (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 $f(x)=\sqrt{x}$ defined on $3$ 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. This is discussed at more length in the article Hölder condition.

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Examples

Lipschitz continuity in metric spaces

Properties of Lipschitz continuous functions

Hölder continuity

See also