Lipschitz-continuous map: Difference between revisions

Latest revision as of 23:40, 24 November 2008

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

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 ==Definition==
-Suppose <math>(X,d_X)</math> and <math>(Y,d_Y)</math> are [[defining ingredient::metric space]]s. In other words, <math>X</math> and <math>Y</math> are sets, and <math>d_X</math> and <math>d_Y</math> are metrics on <math>X</math> and <math>Y</math> respectively. A function <math>f:X \to Y</math> is termed a '''Lipschitz-continuous map''' if there exists a nonnegative real number <math>K</math> such that:
+Suppose <math>(X,d_X)</math> and <math>(Y,d_Y)</math> are [[defining ingredient::metric space]]s. In other words, <math>X</math> and <math>Y</math> are sets, and <math>d_X</math> and <math>d_Y</math> are metrics on <math>X</math> and <math>Y</math> respectively. A function <math>f:X \to Y</math> is termed a '''Lipschitz-continuous map''' or a '''Lipschitz map''' if there exists a nonnegative real number <math>K</math> such that:
 <math>\ \forall \ a,b \in X, d_Y(f(a),f(b)) \le Kd_X(a,b)</math>.