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:

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

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

• Contraction
• Short map

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