In this paper, we mainly research on fractional differentiability of certain continuous functions with fractal dimension one. First, Riemann–Liouville fractional differential of differentiable functions must exist. Then, we prove the existence of Riemann–Liouville fractional differential of continuous functions satisfying the Lipschitz condition, which means that all of their Riemann–Liouville fractional integral of any positive orders in (0, 1) are differentiable. For continuous functions which do not satisfy the Lipschitz condition, we give counterexamples of certain continuous functions whose Riemann–Liouville fractional differential does not exist of certain positive order in (0, 1). Riemann–Liouville fractional differentiability of other one-dimensional continuous functions has also been investigated elementary. Fractional differentiability takes interpretation on physical problems like moving particle and transports through porous and percolation medium with residual memory.

中文翻译：

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连续函数的黎曼-刘维尔分数可微性及其物理插值

在本文中，我们主要研究某些具有分形维数的连续函数的分数可微性。首先，必须存在可微函数的黎曼-刘维尔分数微分。然后，我们证明了满足 Lipschitz 条件的连续函数的 Riemann-Liouville 分数阶微分的存在，这意味着它们的所有正阶的 Riemann-Liouville 分数阶积分(0, 1)是可微的。对于不满足 Lipschitz 条件的连续函数，我们给出了某些 Riemann-Liouville 分数阶微分不存在的特定正阶连续函数的反例(0, 1). 其他一维连续函数的 Riemann-Liouville 分数可微性也已被初步研究。分数可微分解释物理问题，如移动粒子和通过具有残余记忆的多孔和渗滤介质传输。