|
⇤ ← Revision 1 as of 2025-03-27 15:02:22
Size: 858
Comment: Initial commit
|
← Revision 2 as of 2025-09-24 13:39:35 ⇥
Size: 872
Comment: Moving vector pages
|
| Deletions are marked like this. | Additions are marked like this. |
| Line 13: | Line 13: |
| In terms of [[LinearAlgebra/Distance|Euclidean distance]], a function ''f'' is Lipschitz continuous if there is a constant ''K'' such that ''||f(x) - f(y)|| <= K ||x - y||''. ''K'' is referred to as the '''Lipschitz constant'''. | In terms of [[Calculus/Distance#Euclidean_distance|Euclidean distance]], a function ''f'' is Lipschitz continuous if there is a constant ''K'' such that ''||f(x) - f(y)|| <= K ||x - y||''. ''K'' is referred to as the '''Lipschitz constant'''. |
Lipschitz Continuity
Lipschitz continuity is a restriction placed upon a function. A function is Lipschitz continuous if its rate of change is bounded.
Contents
Description
In terms of Euclidean distance, a function f is Lipschitz continuous if there is a constant K such that ||f(x) - f(y)|| <= K ||x - y||. K is referred to as the Lipschitz constant.
Note that this does differ from constraints defined in terms of differentiability. As an example: the absolute value function is Lipschitz continuous, but is not differentiable everywhere.
A function can be locally Lipschitz continuous without being globally so. f(x) = x2 grows at too fast a rate away from the origin, but passes the test near it.