A Function in the Continuous Hölder Class

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Thread starter Euge
Start date Oct 22, 2023
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Continuous Function

In summary, the Continuous Hölder Class is a set of mathematical functions characterized by their smoothness and regularity, with a specific rate of change known as the Hölder exponent. The Hölder exponent is calculated by taking the supremum of the ratio of the function's values at two points over the distance between those points. It determines the smoothness and regularity of a function, with a higher exponent indicating a smoother function. A function belongs to this class if it has a finite Hölder exponent and is locally Hölder continuous. Some real-world applications of this class include the study of fractals, data analysis, and modeling physical phenomena in fields such as physics and engineering.

Oct 22, 2023

Euge

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Let ##0 < \alpha < 1##. Find a necessary and sufficient condition for the function ##f : [0,1] \to \mathbb{R}##, ##f(x) = \sqrt{x}##, to belong to the class ##C^{0,\alpha}([0,1])##.