Analysis MOC

Lipschitz continuity

A function between metric spaces and is Lipschitz continuous iff. there exists such that

for all . #m/def/anal The smallest with this property is called the Lipschitz constant of , so that . As the name implies, a Lipschitz continuous function is also continuous.

A Contraction map is a Lipschitz continuous map with . #m/def/anal

A Lipschitz function is almost continuously differentiable. The measure of the set of points for which the derivative is undefined is zero.

If the derivative of a function is bounded, than it is Lipschitz.

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