关键词: 不可微连续函数, 自相似性, 分维数, 闪电

Abstract: A family of functions, where each function is everywhere continuous but nowhere differentiable, is constructed. Compared to the existing examples of non-differentiable continuous functions, the functions in this family have the following features. 1) Each function in this family has a lightning-like intuitive image and is therefore called a Lightning Function, and the whole family is called Lightning Function Family. 2) It can be intuitively understood why each lightning function is everywhere continuous but nowhere differentiable. 3) Each lightning function is quasi-self-similar in the sense that a local image would become identical with the global image by stretching in the horizontal and vertical directions. 4) Each lightning function can be obtained by infinite number of repetitions of a corresponding Lightning Transformation starting from the straight segment between the points (0, 0) and (1, 1). Fractal properties of Lightning Functions are also discussed based on Mandelbrot's definition of fractal dimension for coastlines.

Key words: non-differentiable continuous functions, self-similar, fractal dimension, lightning