Lipschitz continuity

In mathematics, a function

Lipschitz continuity in metric spaces

The notion of Lipschitz continuity can be extended to arbitrary metric spaces, when the absolute values in the definition is replaced by general distances. A function

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:f : M → N

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between metric spaces M and N is called Lipschitz continuous if there exists a constant

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:K ≥ 0

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such that

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:d(f(x), f(y)) ≤ K d(x, y)

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for all x and y in M, with the smallest such K again being called the Lipschitz constant. Here, d denotes the distance function in the spaces M and N. The two distance functions could be different; the same notation was used because it is clear from the formula which distance is in which space.

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If f : M → N satisfies the condition

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:d(x,y)/K ≤ d(f(x), f(y)) ≤ K d(x, y)

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where K ≥ 1 then f is called a bilipschitz function. Every bilipschitz function is injective. A bilipschitz bijection is the same thing as a Lipschitz bijection whose inverse function is also Lipschitz.

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