Hölder condition

In mathematics, we say that a function satisfies a Hölder condition, or is

α

{\displaystyle \alpha }

-Hölder continuous or simply Hölder continuous, if for a real or complex-valued function

f

{\displaystyle f}

on

d

{\displaystyle d}

-dimensional Euclidean space, i.e.

f

:

Ω

→

R

{\displaystyle f:\Omega \to \mathbb {R} }

or

C

{\displaystyle \mathbb {C} }

(where

Ω

⊆

R

d

{\displaystyle \Omega \subseteq \mathbb {R} ^{d}}

or

C

d

{\displaystyle \mathbb {C} ^{d}}

), when there are real constants

C

≥

0

{\displaystyle C\geq 0}

,

α

>

0

{\displaystyle \alpha >0}

, such that

|

f

(

x

)

−

f

(

y

)

|

≤

C

‖

x

−

y

‖

α

{\displaystyle |f(x)-f(y)|\leq C\|x-y\|^{\alpha }}

for all

x

,

y

∈

Ω

{\displaystyle x,y\in \Omega }

. More generally, the condition can be formulated for functions between any two metric spaces. The number

α

{\displaystyle \alpha }

is called the exponent of the Hölder condition. A function on an interval satisfying the condition with

α

>

1

{\displaystyle \alpha >1}

is constant (see proof below). If

α

=

1

{\displaystyle \alpha =1}

, then the function satisfies a Lipschitz condition. For any

α

>

0

{\displaystyle \alpha >0}

, the condition implies the function is uniformly continuous. The condition is named after Otto Hölder.

If

α

=

0

{\displaystyle \alpha =0}

, the function is simply bounded (any two values

f

{\displaystyle f}

takes are at most

C

{\displaystyle C}

apart).

We have the following chain of inclusions for functions defined on a closed and bounded interval [a, b] of the real line with a < b:

where 0 < α ≤ 1.