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:
 * $\exists K \in \R_{\ge 0}: \forall x, y \in I: \left|{f \left({x}\right) - f \left({y}\right)}\right| \le K \left|{x - y}\right|$

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 $I$.