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 $I$.
Also see
- Results about Lipschitz continuity can be found here.
Sources
- 2013: Francis Clarke: Functional Analysis, Calculus of Variations and Optimal Control ... (previous) ... (next): $1.1$: Basic Definitions