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

摘要： 构造了一族处处连续而又处处不可微的函数。与已有的不可微连续函数相比, 这族函数有如下特点。1)每个函数都有直观的形象???形如闪电, 故称为 “闪电函数 ”, 这族函数之全体称为 “闪电函数族”;2)很容易直观地理解为什么每个闪电函数是处处连续而又处处不可微的; 3)每个闪电函数均具有如下意义下的准自相似性: 其局部图像适当地在横向和纵向拉伸后将与整体图像相同; 4)每个闪电函数均可由无穷次地重复相应的“闪电变换”而获得。根据 Mandelbrot 关于海岸线分维数的定义, 还讨论了闪电函数的分维性。

关键词: 不可微连续函数, 自相似性, 分维数, 闪电