Lipschitz

A bilipschitz function is the same thing as an injective Lipschitz function whose inverse function is also Lipschitz.

A function is called locally Lipschitz continuous if for every x in X there exists a neighborhood U of x such that f restricted to U is Lipschitz continuous.

By the result of the previous paragraph, the limit function is also Lipschitz, with the same bound for the Lipschitz constant.

Equivalently, if X is a locally compact metric space, then f is locally Lipschitz if and only if it is Lipschitz continuous on every compact subset of X. In spaces that are not locally compact, this is a necessary but not a sufficient condition.

It is also true that every real-valued Lipschitz-continuous map defined on a subset of a metric space can be extended to a Lipschitz-continuous map on the whole space.

It is possible that the function F could have a very large Lipschitz constant but a moderately sized, or even negative, one-sided Lipschitz constant.

Gita Lipschitz, Danie Steele, Nic Ingel, Paris Obel, Miya Ichikowitz and Evan Koton at the King David Victory Park addiction awareness talk.

For instance, every function that has bounded first derivatives is Lipschitz.

However, the discrete metric space is free in the category of bounded metric spaces and Lipschitz continuous maps, and it is free in the category of metric spaces bounded by 1 and short maps.

In particular, any continuously differentiable function is locally Lipschitz, as continuous functions are locally bounded so its gradient is locally bounded as well.

More generally, a set of functions with bounded Lipschitz constant forms an equicontinuous set.

Moreover, if K is the best Lipschitz constant of f, then whenever the total derivative Df exists.

This result does not hold for sequences in which the functions may have unbounded Lipschitz constants, however.