We obtain a criterion for an analytic subset of a Euclidean space to contain points of differentiability of a typical Lipschitz function: namely, that it cannot be covered by countably many sets, each of which is closed and purely unrectifiable (has a zero-length intersection with every $C^1$ curve). Surprisingly, we establish that any set failing this criterion witnesses the opposite extreme of typical behaviour: in any such coverable set, a typical Lipschitz function is everywhere severely non-differentiable.

中文翻译：

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通过典型可微性对集合进行二分法

我们获得了欧几里得空间的分析子集包含典型 Lipschitz 函数的可微点的标准：即，它不能被可数的多个集合所覆盖，每个集合都是封闭的且完全不可纠正（具有零长度交集与每 $C^1$ 曲线）。令人惊讶的是，我们发现任何不符合这个标准的集合都见证了典型行为的相反极端：在任何这样的可覆盖集合中，典型的 Lipschitz 函数在任何地方都是严重不可微的。