Lipschitz-continuous map

Definition
Suppose $$(X,d_X)$$ and $$(Y,d_Y)$$ are defining ingredient::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:

$$\ \forall \ a,b \in X, d_Y(f(a),f(b)) \le Kd_X(a,b)$$.

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

Stronger properties

 * Weaker than::Contraction
 * Weaker than::Short map

Weaker properties

 * Uniformly continuous map:
 * Continuous map