x^(a)sin x^(b)型函数在构造反例中的应用

The Application of x^(a)sin x^(b)-type Functions to Counter Examples

利用一类由幂函数与正弦函数构成的初等函数,构造了数学分析中的许多反例,包括函数的有界性问题、单调性问题、连续性问题(含一致连续性)、可微性问题、极值问题及可积性问题(含原函数、Riemann可积性、反常积分)等问题中的反例.

Using a class of elementary functions consisting only of power functions and sine functions,we construct many counter examples in mathematical analysis,which include the counter examples in boundness problem,monotone problem,continuity problem(containing uniform continuity),differentiability problem,extremum problem and integrability problem(containing primitive,Riemannian integrability and improper integral).