The Morse–Sard theorem gives conditions under which the set of critical values of a function between Euclidean spaces has Lebesgue measure zero. Over the years the result has been extended and strengthened in various ways. We present a result, along with a simple proof, that subsumes many of these generalizations. We also review methods of constructing examples showing that differentiability hypotheses cannot be weakened, and we construct a complete set of examples for our result.

中文翻译：

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广义莫尔斯-萨德定理

Morse-Sard 定理给出了欧几里得空间之间函数的临界值集合的勒贝格测度为零的条件。多年来，该成果以各种方式得到扩展和加强。我们提出了一个结果以及一个简单的证明，其中包含了许多这些概括。我们还回顾了构建示例的方法，表明不能削弱可微性假设，并且我们为我们的结果构建了一组完整的示例。