Math 2.0

Real functions Definition

Lipschitz map

Builds on Metric space

Definition

Let (X,dX)(X, d_X) and (Y,dY)(Y, d_Y) be metric spaces, let f ⁣:X→Yf \colon X \to Y be a function and let K≥0K \ge 0 be a real number. The map ff is Lipschitz with constant KK when dY(f(x),f(y))≤K dX(x,y)for all x,y∈X.d_Y\bigl(f(x), f(y)\bigr) \le K \, d_X(x, y) \quad \text{for all } x, y \in X. The map ff is Lipschitz when it is Lipschitz with some constant K≥0K \ge 0.

Remarks

The smallest such KK is the Lipschitz constant of ff. The map x↦∣x∣x \mapsto |x| on ℝ\mathbb{R} is Lipschitz with constant 11. The map x↦xx \mapsto \sqrt{x} on [0,∞)[0, \infty) is not Lipschitz, because x/x\sqrt{x} / x is not bounded near 00. A Lipschitz map with constant K<1K < 1 is a contraction. A differentiable f ⁣:ℝ→ℝf \colon \mathbb{R} \to \mathbb{R} with ∣f′(x)∣≤K|f'(x)| \le K for all xx is Lipschitz with constant KK, by the mean value theorem.

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Further reading

  1. Web https://en.wikipedia.org/wiki/Lipschitz_continuity, Definitions
About this page
Tag0TMZ21WKCN
URIm2://0TMZ21WKCN
Aliaseslipschitz-function
Kinddefinition
WritingW1 stated
MSC26A16
DefinesLipschitz condition
UsesMetric space

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