# Lipschitz condition

A mapping $f:X\to Y$ between metric spaces is said to satisfy the
Lipschitz condition^{}, or to be *Lipschitz continuous* or *$L$-Lipschitz* if there exists a real constant $L$ such
that

$${d}_{Y}(f(p),f(q))\le L{d}_{X}(p,q),\text{for all}p,q\in X.$$ |

The least constant $L$ for which the previous inequality^{} holds, is called the *Lipschitz constant* of $f$.
The space of Lipschitz continuous functions is often denoted by $\mathrm{Lip}(X,Y)$.

Clearly, every Lipschitz continuous function is continuous^{}.

## Notes.

More generally, one says that a mapping satisfies a Lipschitz condition of order $\alpha >0$ if there exists a real constant $C$ such that

$${d}_{Y}(f(p),f(q))\le C{d}_{X}{(p,q)}^{\alpha},\text{for all}p,q\in X.$$ |

Functions which satisfy this condition are also called *Hölder continuous* or *$\alpha $-Hölder*. The vector space of such functions is denoted by ${C}^{0,\alpha}(X,Y)$ and hence $\mathrm{Lip}={C}^{0,1}$.

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 |