Lipschitz continuity

A function $𝑓 : 𝑋 \to 𝑌$ between metric spaces $(𝑋, 𝑑 𝑋)$ and $(𝑌, 𝑑 𝑌)$ is Lipschitz continuous iff. there exists $𝐿$ such that

𝑑 𝑌 (𝑓 (𝑥), 𝑓 (𝑦)) \leq 𝐿 𝑑 𝑋 (𝑥, 𝑦)

for all $𝑥, 𝑦 \in 𝑋$ . #m/def/anal The smallest $𝐿$ with this property is called the Lipschitz constant of $𝑓$ , so that $L i p (𝑓) \leq 𝐿$ . As the name implies, a Lipschitz continuous function is also continuous.

A Contraction map is a Lipschitz continuous map with $0 \leq 𝐿 < 1$ . #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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