Definition:Lipschitz Condition

Metric Space

Let $f$ be a function mapping a metric space $\left({M, d_M}\right)$ to another metric space $\left({N, d_N}\right)$.

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 $M$.

Also see

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

• Lipschitz Equivalence

Source of Name

This entry was named for Rudolf Lipschitz.