Lipschitz continuity

In mathematics, a function

Related Topics:
Mathematics - Function

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:f : I → R

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

defined on an interval of real numbers with real values is called Lipschitz continuous (or is said to satisfy a Lipschitz condition) if there exists a constant

Related Topics:
Interval - Real number

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:K ≥ 0

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

such that

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:|f(x)-f(y)|le K |x-y|

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

for all x, y in the interval I. The smallest such K is called the Lipschitz constant of the map. The name is after the German mathematician Rudolf Lipschitz.

Related Topics:
German - Rudolf Lipschitz

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

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 its Lipschitz constant K. The mean value theorem can be used to prove that any differentiable function with bounded derivative is Lipschitz continuous, with the Lipschitz constant being the largest magnitude of the derivative.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~