特殊高阶函数的构造方法及相关性质证明

Construction Method of Special Higher-order Function andProof of Related Properties

以魏尔施特拉斯理论为基础,基于正、余弦2种三角函数在周期性上的自相似性,通过函数项级数构造出了特殊高阶3种函数:处处连续而处处不可微函数、处处连续而处处非赫尔德连续函数、处处赫尔德连续而不更高阶连续函数,同时证明了这3种函数的相关性质.结果表明:例题证明所提出的构造方法切实有效.

Based on Weilstrass's theory,and the self similarity of sine function and cosine function in periodicity,three special higher-order functions are constructed through function series:everywhere all continuous and everywhere non-differentiable function,everywhere all continuous but not Herder continuous function,everywhere Herder continuous but not higher order continuous function.The related properties of these three functions are proved.An example shows that the construction method proposed in this paper is effective.