Hölder condition

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a real or complex-valued function f on d-dimensional <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean space</a> satisfies a Hölder condition, or is Hölder continuous, when there are real constants C ≥ 0, α > 0, such that

|
        
        f
        (
        x
        )
        −
        f
        (
        y
        )
        
          |
        
        ≤
        C
        ‖
        x
        −
        y
        
          ‖
          
            α
          
        
      
    
    {\displaystyle |f(x)-f(y)|\leq C\|x-y\|^{\alpha }}

for all x and y in the domain of f. More generally, the condition can be formulated for functions between any two <a href="/facts/Metric_space/gnBBRXtc">metric spaces</a>. The number 
 
 
 
 α
 
 
 {\displaystyle \alpha }
 
 is called the exponent of the Hölder condition. A function on an interval satisfying the condition with α > 1 is <a href="/facts/Constant_function/BmPx6dn7">constant</a> (see proof below). If α = 1, then the function satisfies a <a href="/facts/Lipschitz_condition/Hw20EPEU">Lipschitz condition</a>. For any α > 0, the condition implies the function is <a href="/facts/Uniformly_continuous/hjvw7tII">uniformly continuous</a>. The condition is named after <a href="/facts/Otto_H%C3%B6lder/KRMmwcTI">Otto Hölder</a>.
If 
 
 
 
 α
 =
 0
 
 
 {\displaystyle \alpha =0}
 
, the function is simply <a href="/facts/Bounded_function/whVjbtuk">bounded</a> (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:

<a href="/facts/Continuously_differentiable/jQqTgLk1">Continuously differentiable</a> ⊂ <a href="/facts/Lipschitz_continuous/Hw20EPEU">Lipschitz continuous</a> ⊂ 
 
 
 
 α
 
 
 {\displaystyle \alpha }
 
-Hölder continuous ⊂ <a href="/facts/Uniformly_continuous/hjvw7tII">uniformly continuous</a> = <a href="/facts/Continuous_function/sKbl02pB">continuous</a>,
where 0 < α ≤ 1.

Hölder condition open-in-new

Hölder condition