Abstract The purpose of this article is to explore the very general phenomenon that a function between metric spaces has a particular metric property if and only if whenever it is followed in a composition by an arbitrary real-valued Lipschitz function, the composition has this property. The key tools we use are the Efremovic lemma [21] and a theorem of Garrido and Jaramillo [22] that says that a function h between metric spaces is Lipschitz if and only if whenever it is followed by a Lipschitz real-valued function in a composition, the composition is Lipschitz. We also present a streamlined proof of the Garrido-Jaramillo result itself, but one that still relies on their natural continuous linear operator from the Lipschitz space for the target space to the Lipschitz space for the domain. Separately, we include a highly applicable uniform closure theorem that yields the most important uniform density theorems for Lipschitz-type functions as special cases.

中文翻译：

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实值 Lipschitz 函数和函数的度量属性

摘要 本文的目的是探索一个非常普遍的现象，即度量空间之间的函数具有特定的度量属性，当且仅当在组合中跟随任意实值 Lipschitz 函数时，该组合具有该属性。我们使用的关键工具是 Efremovic 引理 [21] 和 Garrido 和 Jaramillo [22] 的定理，该定理表明度量空间之间的函数 h 是 Lipschitz 当且仅当它后面跟着一个 Lipschitz 实值函数成分，成分是Lipschitz。我们还提供了 Garrido-Jaramillo 结果本身的简化证明，但仍然依赖于从 Lipschitz 空间的目标空间到域的 Lipschitz 空间的自然连续线性算子。分别地，