Definition:Lipschitz Continuity

Definition
Let $M = \struct {A, d}$ and $M' = \struct {A', d'}$ be metric spaces.

Let Let $f: A \to A'$ be a mapping.

Then $f$ is a Lipschitz continuous mapping there exists a positive real number $K \in \R_{\ge 0}$ such that:
 * $\forall x, y \in A: \map {d'} {\map f x, \map f y} \le K \map d {x, y}$

That is, the distance between the images of two points lies within a fixed multiple of the distance between the points.

Real Function
The concept can be directly applied to the real numbers considered as a metric space under the usual topology:

Also known as
A Lipschitz continuous mapping $f: A \to A'$ is also seen referred to as follows:


 * $f$ satisfies the Lipschitz condition on $A$
 * $f$ is a Lipschitz function on $A$
 * $f$ is Lipschitz on $A$.

Also see

 * Lipschitz Condition implies Uniform Continuity


 * Definition:Lipschitz Equivalence