# Lipschitz continuity

In mathematical analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exists a definite real number such that, for every pair of points on the graph of this function, the absolute value of the slope of the line connecting them is not greater than this real number; this bound is called the function's "Lipschitz constant" (or "modulus of uniform continuity").

In the theory of differential equations, Lipschitz continuity is the central condition of the Picard–Lindelöf theorem which guarantees the existence and uniqueness of the solution to an initial value problem. A special type of Lipschitz continuity, called contraction, is used in the Banach fixed point theorem.

We have the following chain of inclusions for functions over a compact subset of the real line

Continuously differentiableLipschitz continuousα-Hölder continuousuniformly continuouscontinuous

where 0 < α ≤1. We also have

Lipschitz continuousabsolutely continuousbounded variationdifferentiable almost everywhere

## Definitions For a Lipschitz continuous function, there is a double cone (shown in white) whose vertex can be translated along the graph, so that the graph always remains entirely outside the cone.

Given two metric spaces (X, dX) and (Y, dY), where dX denotes the metric on the set X and dY is the metric on set Y (for example, Y might be the set of real numbers R with the metric dY(x, y) = |xy|, and X might be a subset of R), a function f : XY is called Lipschitz continuous if there exists a real constant K ≥ 0 such that, for all x1 and x2 in X,

$d_{Y}(f(x_{1}),f(x_{2}))\leq Kd_{X}(x_{1},x_{2}).$ Any such K is referred to as a Lipschitz constant for the function f. The smallest constant is sometimes called the (best) Lipschitz constant; however in most cases the latter notion is less relevant. If K = 1 the function is called a short map, and if 0 ≤ K < 1 the function is called a contraction.

The inequality is (trivially) satisfied if x1 = x2. Otherwise, one can equivalently define a function to be Lipschitz continuous if and only if there exists a constant K ≥ 0 such that, for all x1x2,

${\frac {d_{Y}(f(x_{1}),f(x_{2}))}{d_{X}(x_{1},x_{2})}}\leq K.$ For real-valued functions of several real variables, this holds if and only if the absolute value of the slopes of all secant lines are bounded by K. The set of lines of slope K passing through a point on the graph of the function forms a circular cone, and a function is Lipschitz if and only if the graph of the function everywhere lies completely outside of this cone (see figure).

A function is called locally Lipschitz continuous if for every x in X there exists a neighborhood U of x such that f restricted to U is Lipschitz continuous. Equivalently, if X is a locally compact metric space, then f is locally Lipschitz if and only if it is Lipschitz continuous on every compact subset of X. In spaces that are not locally compact, this is a necessary but not a sufficient condition.

More generally, a function f defined on X is said to be Hölder continuous or to satisfy a Hölder condition of order α > 0 on X if there exists a constant M > 0 such that

$d_{Y}(f(x),f(y))\leq Md_{X}(x,y)^{\alpha }$ for all x and y in X. Sometimes a Hölder condition of order α is also called a uniform Lipschitz condition of order α > 0.

If there exists a K ≥ 1 with

${\frac {1}{K}}d_{X}(x_{1},x_{2})\leq d_{Y}(f(x_{1}),f(x_{2}))\leq Kd_{X}(x_{1},x_{2})$ then f is called bilipschitz (also written bi-Lipschitz). A bilipschitz mapping is injective, and is in fact a homeomorphism onto its image. A bilipschitz function is the same thing as an injective Lipschitz function whose inverse function is also Lipschitz. Surjective bilipschitz functions are exactly the isomorphisms of metric spaces.

## Examples

Lipschitz continuous functions
• The function f(x) = Template:Radic defined for all real numbers is Lipschitz continuous with the Lipschitz constant K = 1, because it is everywhere differentiable and the absolute value of the derivative is bounded above by 1. See the first property listed below under "Properties".
• Likewise, the sine function is Lipschitz continuous because its derivative, the cosine function, is bounded above by 1 in absolute value.
• The function f(x) = |x| defined on the reals is Lipschitz continuous with the Lipschitz constant equal to 1, by the reverse triangle inequality. This is an example of a Lipschitz continuous function that is not differentiable. More generally, a norm on a vector space is Lipschitz continuous with respect to the associated metric, with the Lipschitz constant equal to 1.
Lipschitz continuous functions that are not everywhere differentiable
• The function f(x) = |x|
Continuous functions that are not (globally) Lipschitz continuous
• The function f(x) = Template:Radic defined on [0, 1] is not Lipschitz continuous. This function becomes infinitely steep as x approaches 0 since its derivative becomes infinite. However, it is uniformly continuous as well as Hölder continuous of class C0, α for α ≤ 1/2.
Differentiable functions that are not (globally) Lipschitz continuous
• The function f(x) = x3/2sin(1/x) where x ≠ 0 and f(0) = 0, restricted on [0, 1], gives an example of a function that is differentiable on a compact set while not locally Lipschitz because its derivative function is not bounded. See also the first property below.
Analytic functions that are not (globally) Lipschitz continuous
• The exponential function becomes arbitrarily steep as x → ∞, and therefore is not globally Lipschitz continuous, despite being an analytic function.
• The function f(x) = x2 with domain all real numbers is not Lipschitz continuous. This function becomes arbitrarily steep as x approaches infinity. It is however locally Lipschitz continuous.

## Properties

• An everywhere differentiable function g : R → R is Lipschitz continuous (with K = sup |g′(x)|) if and only if it has bounded first derivative; one direction follows from the mean value theorem. In particular, any continuously differentiable function is locally Lipschitz, as continuous functions are locally bounded so its gradient is locally bounded as well.
${\tilde {f}}(x):=\inf _{u\in U}\{f(u)+k\,d(x,u)\},$ where k is a Lipschitz constant for f on U.

## Lipschitz manifolds

Let U and V be two open sets in Rn. A function T : UV is called bi-Lipschitz if it is a Lipschitz homeomorphism onto its image, and its inverse is also Lipschitz.

Using bi-Lipschitz mappings, it is possible to define a Lipschitz structure on a topological manifold, since there is a pseudogroup structure on bi-Lipschitz homeomorphisms. This structure is intermediate between that of a piecewise-linear manifold and a smooth manifold. In fact a PL structure gives rise to a unique Lipschitz structure; it can in that sense 'nearly' be smoothed.

## One-sided Lipschitz

Let F(x) be an upper semi-continuous function of x, and that F(x) is a closed, convex set for all x. Then F is one-sided Lipschitz if

$(x_{1}-x_{2})^{T}(F(x_{1})-F(x_{2}))\leq C\Vert x_{1}-x_{2}\Vert ^{2}$ for some C for all x1 and x2.

It is possible that the function F could have a very large Lipschitz constant but a moderately sized, or even negative, one-sided Lipschitz constant. For example the function

${\begin{cases}F:\mathbf {R} ^{2}\to \mathbf {R} ,\\F(x,y)=-50(y-\cos(x))\end{cases}}$ has Lipschitz constant K = 50 and a one-sided Lipschitz constant C = 0. An example which is one-sided Lipschitz but not Lipschitz continuous is $F(x)=e^{-x}$ , with C = 0.