# Definition:Lipschitz Continuity/Real Function

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## 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 $I$**.

## Sources

- 2013: Francis Clarke:
*Functional Analysis, Calculus of Variations and Optimal Control*... (previous) ... (next): $1.1$: Basic Definitions