函数的连续性、不可微性与自相似性方法

Continuity, Non-Differentiability and Self-Similarity of Functions

利用正弦函数和余弦函数的自相似性,运用傅里叶级数的理论,给出处处连续但处处不可微;处处连续但处处不赫尔德连续;处处赫尔德连续但又处处不更高阶连续的函数的构造方法,并对这类函数的相关性质给出严格的证明.通过实例,说明了这种构造方法的可行性.

By using the self-similarity of sine function and cosine function and the theory of Fourier series, this paper gives the construction methods of functions which are continuous everywhere but non-differentiable everywhere, continuous everywhere but not H lder continuous everywhere, continuous everywhere but not higher order continuous everywhere, and gives a strict proof of the related properties of these functions. Some examples show that this construction method is feasible.