Math 2.0

Real functions Definition

Hölder continuity

Builds on Metric spaceLipschitz map

Definition

Let (X,dX)(X, d_X) and (Y,dY)(Y, d_Y) be metric spaces, let f ⁣:X→Yf \colon X \to Y be a function, and let C≥0C \ge 0 and α>0\alpha > 0 be real numbers. The map ff is Hölder continuous with constant CC and exponent α\alpha when dY(f(x),f(y))≤C dX(x,y)αfor all x,y∈X.d_Y\bigl(f(x), f(y)\bigr) \le C \, d_X(x, y)^{\alpha} \quad \text{for all } x, y \in X.

Remarks

For α=1\alpha = 1, this is the condition for a Lipschitz map with constant CC. The map x↦xx \mapsto \sqrt{x} on [0,∞)[0, \infty) is Hölder continuous with C=1C = 1 and α=1/2\alpha = 1/2. A Hölder continuous map is uniformly continuous: for ε>0\varepsilon > 0 and C>0C > 0, take δ=(ε/C)1/α\delta = (\varepsilon / C)^{1/\alpha}. For a real function on an interval, Hölder continuity with an exponent α>1\alpha > 1 forces the function to be constant.

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Element, definition, no evidence levelElementdefinitionProposition, concept, no evidence levelPropositionconceptReal numbers, object, no evidence levelReal numbersobjectSet, definition, no evidence levelSetdefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionFunction, definition, no evidence levelFunctiondefinitionMetric space, definition, no evidence levelMetric spacedefinitionLipschitz map, definition, no evidence levelLipschitz mapdefinitionHölder continuity, definition, no evidence levelHölder continuitydefinition
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  16. Lipschitz map definition
  17. Hölder continuity definition

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Further reading

  1. Web https://en.wikipedia.org/wiki/H%C3%B6lder_condition, lead section
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Kinddefinition
WritingW1 stated
MSC26A16
DefinesHölder continuity
UsesMetric space, Lipschitz map

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