关于函数严格单调性的一个注记

A Note on the Rigid Monotony of Functions

对于几乎可导的连续函数,给出了严格单调的一个充要条件,证明了定义在区间D上的几乎可导函数f(x)严格递增(严格递减)当且仅当f’(x)在D上几乎非负(几乎非正),且D^+(D^-)是D的调子集,其中D^+={x∈D:f'(x)>0}(D^-={x∈D:f'(x)<0})。这改进了有关函数严格单调性的一些结果。

In this paper, the author gives an essential condition of rigid monotony to the almost derivable functions. It proves that an almost derivable functionf (x ) on an interval D is stricdy increasing ( strictly decreasing) if and only if f' (x) is almost nonnegative (almost non-positive) on D and D(D) is dense subset of D, where D+ =(x ∈D:f'(x)>0} (D ={x∈ D:f' (x) <0}. This improves some results on rigid monotony of functions.