Hölder condition
Type of continuity of a complex-valued function / From Wikipedia, the free encyclopedia
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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, > 0, such that
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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 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.
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 ⊂ -Hölder continuous ⊂ uniformly continuous ⊂ continuous,
where 0 < α ≤ 1.