Van der Waerden函数的极值性及其应用

On Extreme Property of Van der Waerden Function and Its Application

通过对Van der Waerden函数的讨论,证明Van der Waerden函数在所有的有限小数处取得极小值.给出了连续函数可以有无穷多但至多可数个极值点,并且这些极值点可以在定义域内稠密的结果.在推广了相关文献已有结论的同时,从极值分布的角度考察了常见的处处连续但处处不可导函数的相关特征.

The property of Van der Waerden function was discussed in this paper and it is proved that the decimal fractions of limit digits are all the local minimum points of Van der Waerden function. It is deduced that the local extreme points of a continuous function can be infinite and at most denumerable, and the local extreme points can be dense in the domain of the continuous function. Thus the present result in the bibliography was ex- tended, and the character of the function which is continuous everywhere and derivable nowhere was illustrate in a new way.