Definition:Lipschitz Continuity/Real Function

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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 $I$.