In this paper, we mainly investigate the fractional differential of a class of continuous functions. The upper Box dimension of the Marchaud fractional differential of continuous functions satisfying the Hölder condition increases at most linearly with the order of the fractional differential when they exist. Furthermore, if a continuous function satisfies the Lipschitz condition, the upper Box dimension of its Marchaud fractional differential is at most the sum of one and order of the fractional differential when it exists. From the point of view of the fractal dimension, it increases at most linearly with the fractional order.

中文翻译：

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某些连续函数的马绍分数微分的分形维数估计

本文主要研究一类连续函数的分数阶微分。满足 Hölder 条件的连续函数的 Marchaud 分数阶微分的上 Box 维数在存在时最多随分数阶微分的阶数线性增加。此外，如果一个连续函数满足 Lipschitz 条件，则其 Marchaud 分数阶微分的上 Box 维数最多为分数阶微分存在时的 1 和阶之和。从分形维数的角度来看，它最多随分数阶线性增加。