Definition:Lipschitz Continuity

Definition
Let $M = \left({A, d}\right)$ and $M' = \left({A', d\,'}\right)$ be metric spaces.

Let $f: A \to A'$ be a mapping such that there exists a positive real number $K \in \R_{\ge 0}$ such that:
 * $\forall x, y \in A: d\,' \left({f \left({x}\right), f \left({y}\right)}\right) \le K d \left({x, y}\right)$

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

Then $f$ is a Lipschitz continuous mapping.

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


 * Lipschitz Equivalence