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huldercondition

hulder condition

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Hulder condition might refer to
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 nonnegative real constants C, 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 metric spaces. The number α is called the exponent of the Hölder condition. A function on an interval satisfying the condition with α > 1 is constant. 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 strict inclusions for functions over a closed and bounded non-trivial interval of the real line * * Continuously differentiable ⊂ Lipschitz continuous ⊂ α-Hölder continuous ⊂ uniformly continuous = continuouswhere 0 < α ≤ 1. * *

Hulder condition might refer to

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 nonnegative real constants C, α>0, such that*
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* {\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 α is called the exponent of the Hölder condition. A function on an interval satisfying the condition with α > 1 is constant. 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 strict inclusions for functions over a closed and bounded non-trivial interval of the real line
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* Continuously differentiable ⊂ Lipschitz continuous ⊂ α-Hölder continuous ⊂ uniformly continuous = continuouswhere 0 < α ≤ 1.
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huldercondition