Riemann's non-differentiable function is a classic example of a continuous function which is almost nowhere differentiable, and many results concerning its analytic regularity have been shown so far. However, it can also be given a geometric interpretation, so questions on its geometric regularity arise. This point of view is developed in the context of the evolution of vortex filaments, modelled by the Vortex Filament Equation or the binormal flow, in which a generalisation of Riemann's function to the complex plane can be regarded as the trajectory of a particle. The objective of this document is to show that the trajectory represented by its image does not have a tangent anywhere. For that, we discuss several concepts of tangent vectors in view of the set's irregularity.

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黎曼不可微函数的几何可微性

黎曼不可微函数是一个几乎无处可微的连续函数的经典例子，迄今为止已经显示了许多关于其解析正则性的结果。但是，它也可以给出几何解释，因此出现了有关其几何规律性的问题。这种观点是在涡丝演化的背景下发展起来的，由涡丝方程或副法向流建模，其中黎曼函数对复平面的推广可以被视为粒子的轨迹。本文档的目的是表明其图像表示的轨迹在任何地方都没有切线。为此，考虑到集合的不规则性，我们讨论了切向量的几个概念。