Abstract One of the simplest constructions of a differentiable monster—a function from a nontrivial interval J ⊆ R into R that is everywhere differentiable but monotone on no interval—is as a difference of two strictly increasing differentiable functions, each with its derivative vanishing on a dense subset of its domain. The goal of this work is to characterize differentiable monsters that can be represented in such a “nice” way as those that are a difference of two increasing everywhere differentiable functions. We show that there are differentiable monsters that are not of bounded variation, so they clearly do not admit such “nice” representation. On the other hand, every differentiable monster f : J → R contains many restrictions that admit “nice” representation: for every non-empty open U ⊆ J , there is a non-trivial interval I ⊆ U such that f ↾ I is a difference of two increasing differentiable maps, each with its derivative vanishing on a dense subset of I.

中文翻译：

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无处单调可微函数和有界变异

摘要 可微怪物的最简单构造之一——从非平凡区间 J ⊆ R 到 R 处处可微但在无区间上单调的函数——是两个严格递增的可微函数的差，每个函数的导数在其域的密集子集。这项工作的目标是描述可微怪物的特征，这些怪物可以以“很好”的方式表示为两个递增的处处可微函数的差异。我们表明存在不具有有限变异的可微怪物，因此它们显然不承认这种“漂亮”的表示。另一方面，每个可微的怪物 f : J → R 包含许多允许“漂亮”表示的限制：对于每个非空的开放 U ⊆ J ，