Lipschitz continuity

In mathematics, a function

Properties of Lipschitz continuous functions

Every Lipschitz continuous map is uniformly continuous, and hence a fortiori continuous.

Related Topics:
Uniformly continuous - Continuous

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Lipschitz continuous maps with Lipschitz constant K = 1 are called short maps and with

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K < 1 are called contraction mappings if M=N; the latter are the subject of the Banach fixed point theorem.

Related Topics:
Contraction mapping - Banach fixed point theorem

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Lipschitz continuity is an important condition in the existence and uniqueness theorem for ordinary differential equations.

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If U is a subset of the metric space M and f : U → R is a Lipschitz continuous map, there always exist Lipschitz continuous maps M → R which extend f and have the same Lipschitz constant as f (see also Kirszbraun theorem).

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A Lipschitz continuous map f : I → R, where I is an interval in R, is almost everywhere differentiable (everywhere except on a set of Lebesgue measure 0). If K is the Lipschitz constant of f, then |f'(x)| ≤ K whenever the derivative exists. Conversely, if f : I → R is a differentiable map with bounded derivative, |f'(x)| ≤ L for all x in I, then f is Lipschitz continuous with Lipschitz constant K ≤ L, a consequence of the mean value theorem.

Related Topics:
Interval - Almost everywhere - Differentiable - Lebesgue measure - Mean value theorem

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All Banach spaces have the notion of Lipschitz continuity.

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