论文标题
关于变异功能及其连续性的模量
On variation functions and their moduli of continuity
论文作者
论文摘要
我们研究了有界变异及其变异功能的功能连续性的模量。很容易看出，有界变化函数的连续性模量总是更小或等于其变异函数的连续性模量。我们表明，如果我们只知道父函数本身的连续性模量，就无法对变异函数连续性的模量进行任何合理的结论。特别是，给定两个连续性模量，第一个比Lipschitz的连续性弱，我们表明存在有界变化的函数，最小的连续性模量小于连续性的第一个模量，但具有最小的连续模量的变化功能，其连续性的最小模量大于第二个连续性模量。特别是，这是否定的问题，无论是$α$-Hölder连续函数的变化功能是否为$α$-Hölder连续，都可以消极地解决问题。
We study the moduli of continuity of functions of bounded variation and of their variation functions. It is easy to see that the modulus of continuity of a function of bounded variation is always smaller or equal to the modulus of continuity of its variation function. We show that we cannot make any reasonable conclusion on the modulus of continuity of the variation function if we only know the modulus of continuity of the parent function itself. In particular, given two moduli of continuity, the first being weaker than Lipschitz continuity, we show that there exists a function of bounded variation with minimal modulus of continuity less than the first modulus of continuity, but with a variation function with minimal modulus of continuity greater than the second modulus of continuity. In particular, this negatively resolves the open problem whether the variation function of an $α$-Hölder continuous function is $α$-Hölder continuous.
