关于凸函数定义的几点思考

考察十余种国内通用的高等数学教材对凸函数定义的异同,论证凸函数的两种原始定义之间的关系,给出凸函数所满足的三个重要不等式,并证明凸函数在一元函数微分学范围内所满足的两个重要性质——连续性和单侧可导性。利用实分析中的Hausdorff极大定理给出满足第一种原始定义而不满足第二种原始定义的一个函数的例子．

Note on the Definition of Convex Function

The definitions of convex function in some native college mathematics textbooks are investigated. The relationship between the two original definitions of convex function is explored. Then three important inequalities about convex function are presented. With these inequalities, it gets that a convex function is of continuity and unilateral derivability. With Hausdorff maximal theorem in real analysis, a function is constructed, which meets the first original definition but doesn＇t do the second.