Hölder condition

In mathematics, we say that a function satisfies a Hölder condition, or is ${\displaystyle \alpha }$-Hölder continuous or simply Hölder continuous, if for a real or complex-valued function ${\displaystyle f}$ on ${\displaystyle d}$-dimensional Euclidean space, i.e. ${\displaystyle f:\Omega \to \mathbb {R} }$ or ${\displaystyle \mathbb {C} }$ (where ${\displaystyle \Omega \subseteq \mathbb {R} ^{d}}$ or ${\displaystyle \mathbb {C} ^{d}}$), when there are real constants ${\displaystyle C\geq 0}$, ${\displaystyle \alpha >0}$, such that $${\displaystyle |f(x)-f(y)|\leq C\|x-y\|^{\alpha }}$$ for all ${\displaystyle x,y\in \Omega }$. More generally, the condition can be formulated for functions between any two metric spaces. The number ${\displaystyle \alpha }$ is called the exponent of the Hölder condition. A function on an interval satisfying the condition with ${\displaystyle \alpha >1}$ is constant (see proof below). If ${\displaystyle \alpha =1}$, then the function satisfies a Lipschitz condition. For any ${\displaystyle \alpha >0}$, the condition implies the function is uniformly continuous. The condition is named after Otto Hölder. If ${\displaystyle \alpha =0}$, the function is simply bounded (any two values ${\displaystyle f}$ takes are at most ${\displaystyle C}$ 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:

Lipschitz continuous ⊂ ${\displaystyle \alpha }$-Hölder continuous ⊂ uniformly continuous ⊂ continuous,

where 0 < α ≤ 1.