Lipschitz Continuity - Definitions

Definitions

Given two metric spaces (X, d_X) and (Y, d_Y), where d_X denotes the metric on the set X and d_Y is the metric on set Y (for example, Y might be the set of real numbers R with the metric d_Y(x, y) = |x − y|, and X might be a subset of R), a function

is called Lipschitz continuous if there exists a real constant K ≥ 0 such that, for all x₁ and x₂ in X,

Any such K is referred to as a Lipschitz constant for the function ƒ. 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 x₁ ≠ x₂,

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 ƒ 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

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

then ƒ 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.