Definition:Lipschitz Condition

Metric Space
Let $$f$$ be a function mapping a metric space $$(M, d_M)$$ to another metric space $$(N, d_N)$$.

Then $$f$$ satisfies the Lipschitz condition on $$M$$ if:


 * $$\exists A \in \R, A \ge 0: d_N \left({f \left({x}\right), f \left({y}\right)}\right) \le A d_M \left({x, y}\right)$$

for each $$x, y\in M$$.

The smallest such $$A$$ is known as the Lipschitz constant of $$f$$.

Real Number Line
Let $$f$$ be a real function.

Let $$I \subseteq \R$$ be a real interval on which:
 * $$\exists A \in \R: \forall y_1, y_2 \in I: \left|{f \left({y_1}\right) - f \left({y_2}\right)}\right| \le A \left|{y_1 - y_2}\right|$$.

Then $$f$$ satisfies the Lipschitz condition in $$I$$.

Alternative terminology
$$f$$ satisfies the Lipschitz condition on $$M$$ can also be worded:
 * $$f$$ is a Lipschitz function on $$M$$;
 * $$f$$ is Lipschitz on $$M$$.

Properties
Every Lipschitz function on $$M$$ is uniformly continuous. (See Lipschitz Condition Implies Uniform Continuity.)

Also see

 * Lipschitz Equivalence