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A mapping $f: X \to Y$ between metric spaces is said to satisfy the Lipschitz condition, or to be Lipschitz continuous or $L$ -Lipschitz if there exists a real constant $L$ such that $$ d_Y(f(p),f(q)) \leq L d_X(p,q),\quad \text{for all}\; p,q\in X.$$
The least constant $L$ for which the previous inequality holds, is called the Lipschitz constant of $f$ . The space of Lipschitz continuous functions is often denoted by $\mathrm{Lip}(X,Y)$ .
Clearly, every Lipschitz continuous function is continuous.
More generally, one says that a mapping satisfies a Lipschitz condition of order $\alpha>0$ if there exists a real constant $C$ such that $$ d_Y(f(p),f(q)) \leq C d_X(p,q)^\alpha,\quad \text{for all}\; p,q\in X.$$
Functions which satisfy this condition are also called Hölder continuous or $\alpha$ -Hölder. The vector space of such functions is denoted by $C^{0,\alpha}(X,Y)$ and hence $\mathrm{Lip}=C^{0,1}$ .
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