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In mathematics, a real-valued function f on Rⁿ satisfies a Hölder condition, or is Hölder continuous, when there are nonnegative real constants C, α, such that, <math> \forall x, y \in \mathbf{R}^n </math>,

<math> | f(x) - f(y) | \leq C |x - y| ^{\alpha}. </math>

This condition generalizes to functions between any two metric spaces. The number α is called the exponent of the Hölder condition. If <math>\alpha=1</math>, then the function satisfies a Lipschitz condition. If <math>\alpha=0</math>, then the function simply is bounded.

Hölder spaces consisting of functions satisfying a Hölder condition are basic in areas of functional analysis relevant to solving partial differential equations. The Hölder space <math> C^{n, \alpha} (\Omega)</math>, where Ω is an open subset of some euclidean space, consists of those functions whose derivatives up to order n are Hölder continuous with exponent α. This is a topological vector space, where

<math> \| f \|_{C^{0,\alpha}} = \sup_{x,y \in \Omega} \frac{| f(x) - f(y) |}{|x-y|^\alpha}, </math>

and for <math> n>0 </math> the norm is given by

<math> \| f \|_{C^{n, \alpha}} = \sum_{| \beta | \leq n} \| D^\beta f \|_{C^{0,\alpha}}</math>

where β ranges over multi-indices.

[edit] Examples in <math>C^{0,\alpha}({\mathbb R})</math>

• If <math>0<\alpha\leq\beta\leq1</math> then all <math>C^{0,\beta}</math> Hölder continuous functions are also <math>C^{0,\alpha}</math> Hölder continuous. This also includes <math>\beta=1</math> and therefore all Lipschitz continuous functions are also <math>C^{0,\alpha}</math> Hölder continuous.

• The function <math>f(x)=\sqrt{x}</math> defined on <math>[0,3]</math> is not Lipschitz continuous, but is <math>C^{0,\alpha}</math> Hölder continuous for <math>\alpha\le\frac12</math>.

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