# Lipschitz condition

A restriction on the behaviour of increase of a function. If for any points and belonging to an interval the increase of a function satisfies the inequality

(*) |

where and is a constant, then one says that satisfies a Lipschitz condition of order on and writes , or . Every function that satisfies a Lipschitz condition with some on is uniformly continuous on , and functions that satisfy a Lipschitz condition of order are absolutely continuous (cf. Absolute continuity; Uniform continuity). A function that has a bounded derivative on satisfies a Lipschitz condition on with any .

The Lipschitz condition (*) is equivalent to the condition

where is the modulus of continuity (cf. Continuity, modulus of) of on . Lipschitz conditions were first considered by R. Lipschitz [1] as a sufficient condition for the convergence of the Fourier series of . In the case the condition (*) is also called a Hölder condition of order .

#### References

[1] | R. Lipschitz, "De explicatione per series trigonometricas insttuenda functionum unius variablis arbitrariarum, et praecipue earum, quae per variablis spatium finitum valorum maximorum et minimorum numerum habent infintum disquisitio" J. Reine Angew. Math. , 63 (1864) pp. 296–308 |

[2] | A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988) |

[3] | I.P. Natanson, "Constructive function theory" , 1–3 , F. Ungar (1964–1965) (Translated from Russian) |

**How to Cite This Entry:**

Lipschitz condition.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Lipschitz_condition&oldid=14093