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 if and only if 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.
At a Point
Let $a \in A$.
$f$ is a Lipschitz continuous at $a$ if and only if there exists a positive real number $K \in \R_{\ge 0}$ such that:
- $\forall x \in A: \map {d'} {\map f x, \map f a} \le K \map d {x, a}$
Lipschitz Constant
Let $f: A \to A'$ be a (Lipschitz continuous) mapping such that:
- $\forall x, y \in A: \map {d'} {\map f x, \map f y} \le K \map d {x, y}$
where $K \in \R_{\ge 0}$ is a positive real number.
Then $K$ is a Lipschitz constant for $f$.
Real Function
The concept can be directly applied to the real numbers considered as a metric space under the usual topology:
Let $A \subseteq \R$.
Let $f: A \to \R$ be a real function.
Let $I \subseteq A$ be a real interval on which:
- $\exists K \in \R_{\ge 0}: \forall x, y \in I: \size {\map f x - \map f y} \le K \size {x - y}$
Then $f$ is Lipschitz continuous on $I$.
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
- Results about Lipschitz continuity can be found here.
Source of Name
This entry was named for Rudolf Otto Sigismund Lipschitz.
Sources
- 2013: Francis Clarke: Functional Analysis, Calculus of Variations and Optimal Control: $2.3$: Convex functions
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Lipschitz condition