Lipschitz functions are uniformly continuous

Builds on Lipschitz continuity Uniform continuity

Statement

Let $D \subseteq \mathbb{R}$ and let $f \colon D \to \mathbb{R}$ be Lipschitz. Then $f$ is uniformly continuous.

Remarks

Let $K > 0$ be a Lipschitz constant. Given $\varepsilon > 0$ , take $\delta = \varepsilon / K$ . The converse fails. The function $\sqrt{x}$ is uniformly continuous on $[0, 1]$ but not Lipschitz. Mathlib states it as LipschitzWith.uniformContinuous.

Reading path

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Limit point of a set definition
Lipschitz continuity definition
Limit of a function definition
Continuity at a point definition
Epsilon-delta definition of continuity statement
Uniform continuity definition
Lipschitz functions are uniformly continuous statement

Tag	`0K1EWAR6AF`
URI	`m2://0K1EWAR6AF`
Kind	statement, proposition
Writing	W0 stub
MSC	26A16
Status	This copy of the site has no build data. Run `m2 status 0K1EWAR6AF`.
Uses	Lipschitz continuity, Uniform continuity

Lipschitz functions are uniformly continuous

Statement

Remarks

Reading path

Further reading

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