Math 2.0

statement proposition

Lipschitz functions are uniformly continuous

Statement

Let D⊆RD \subseteq \mathbb{R} and let f ⁣:D→Rf \colon D \to \mathbb{R} be Lipschitz. Then ff is uniformly continuous.

Remarks

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

Explanations

L1 human

A Lipschitz bound controls the change of ff in the same way at every point. So one input tolerance works for the whole domain.

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Limit point of a set, definition, no evidence levelLimit point of a setdefinitionLipschitz continuity, definition, no evidence levelLipschitz continuitydefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionEpsilon-delta definition of continuity, statement, no evidence levelEpsilon-delta definit…statementUniform continuity, definition, no evidence levelUniform continuitydefinitionLipschitz functions are uniformly continuous, statement, no evidence levelLipschitz functions a…statement
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  1. Limit point of a set definition
  2. Lipschitz continuity definition
  3. Limit of a function definition
  4. Continuity at a point definition
  5. Epsilon-delta definition of continuity statement
  6. Uniform continuity definition
  7. Lipschitz functions are uniformly continuous statement

Sources

  1. Web https://en.wikipedia.org/wiki/Lipschitz_continuity, Properties

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