Wednesday, May 21, 2008

 

Lipschitz continuity

From Wikipedia:

...Lipschitz continuity, named after Rudolf Lipschitz, is a smoothness condition for functions which is stronger than regular continuity. Intuitively, a Lipschitz continuous function is limited in how fast it can change; a line joining any two points on the graph of this function will never have a slope steeper than a certain number called the Lipschitz constant of the function....

* The function f(x) = x^2 with domain all real numbers is not Lipschitz continuous. This function becomes arbitrarily steep as x goes to infinity. It is however locally Lipschitz continuous.

* The function f(x) = x^2 defined on [ − 3,7] is Lipschitz continuous, with Lipschitz constant K = 14.

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