Analysis MOC

Lipschitz continuity

A function 𝑓 :𝑋 →𝑌 between metric spaces (𝑋,𝑑𝑋) and (𝑌,𝑑𝑌) is Lipschitz continuous iff. there exists 𝐿 such that

𝑑𝑌(𝑓(𝑥),𝑓(𝑦))≤𝐿𝑑𝑋(𝑥,𝑦)

for all 𝑥,𝑦 ∈𝑋. anal The smallest 𝐿 with this property is called the Lipschitz constant of 𝑓, so that Lip(𝑓) ≤𝐿. As the name implies, a Lipschitz continuous function is also continuous.

A Contraction map is a Lipschitz continuous map with 0 ≤𝐿 <1. 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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