Lipschitz-continuous map

Definition

Suppose $(X, d_{X})$ and $(Y, d_{Y})$ are metric spaces. In other words, $X$ and $Y$ are sets, and $d_{X}$ and $d_{Y}$ are metrics on $X$ and $Y$ respectively. A function $f : X \to Y$ is termed a Lipschitz-continuous map or a Lipschitz map if there exists a nonnegative real number $K$ such that:

$\forall a, b \in X, d_{Y} (f (a), f (b)) \leq K d_{X} (a, b)$ .

Such a real number $K$ is termed a Lipschitz constant for $f$ . Note that if $K$ is a Lipschitz constant, so is any $L \geq K$ . A function with Lipschitz constant $K = 1$ is termed a short map, while a function with Lipschitz constant $K < 1$ is termed a contraction.

Relation with other properties

Stronger properties

Weaker properties

Uniformly continuous map: For proof of the implication, refer Lipschitz-continuous implies uniformly continuous and for proof of its strictness (i.e. the reverse implication being false) refer Uniformly continuous not implies Lipschitz-continuous
Continuous map