# Definition:Lipschitz Continuity/Real Function

## Definition

Let $A \subseteq \R$.

Let $f: A \to \R$ be a real function.

Let $I \subseteq A$ be a real interval on which:

$\exists K \in \R_{\ge 0}: \forall x, y \in I: \size {\map f x - \map f y} \le K \size {x - y}$

Then $f$ is Lipschitz continuous on $I$.

The constant $K$ is known as a Lipschitz constant for $f$.

## Also known as

A Lipschitz continuous function $f$ is also seen referred to as follows:

• $f$ satisfies the Lipschitz condition on $I$
• $f$ is a Lipschitz function on $I$
• $f$ is Lipschitz on $I$.