Hölder condition

In mathematics, a real or complex-valued function f on d-dimensional Euclidean space satisfies a Hölder condition, or is Hölder continuous, when there are real constants C ≥ 0, α > 0, such that $${\displaystyle |f(x)-f(y)|\leq C\|x-y\|^{\alpha }}$$ for all x and y in the domain of f. More generally, the condition can be formulated for functions between any two metric spaces. The number ${\displaystyle \alpha }$ is called the exponent of the Hölder condition. A function on an interval satisfying the condition with α > 1 is constant (see proof below). If α = 1, then the function satisfies a Lipschitz condition. For any α > 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:

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

where 0 < α ≤ 1.