Lipschitz condition
A mapping between metric spaces is said to satisfy the Lipschitz condition, or to be Lipschitz continuous or -Lipschitz if there exists a real constant such that
The least constant for which the previous inequality holds, is called the Lipschitz constant of . The space of Lipschitz continuous functions is often denoted by .
Clearly, every Lipschitz continuous function is continuous.
Notes.
More generally, one says that a mapping satisfies a Lipschitz condition of order if there exists a real constant such that
Functions which satisfy this condition are also called Hölder continuous or -Hölder. The vector space of such functions is denoted by and hence .
Title | Lipschitz condition |
Canonical name | LipschitzCondition |
Date of creation | 2013-03-22 11:57:48 |
Last modified on | 2013-03-22 11:57:48 |
Owner | paolini (1187) |
Last modified by | paolini (1187) |
Numerical id | 27 |
Author | paolini (1187) |
Entry type | Definition |
Classification | msc 26A16 |
Synonym | Lipschitz |
Synonym | Lipschitz continuous |
Related topic | RademachersTheorem |
Related topic | NewtonsMethod |
Related topic | KantorovitchsTheorem |
Defines | Holder |
Defines | Holder continuous |
Defines | Lipschitz constant |