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 if there exists a nonnegative real number $K$ such that:

$\mathrm{\forall }a,b\in X,d_Y(f(a),f(b))\le K{d}_X(a,b)$.

Such a real number $K$ is termed a Lipschitz constant for $f$. Note that if $K$ is a Lipschitz constant, so is any $L\ge K$. 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