Abstract The notion of weakly monotone functions extends the classical definition of monotone function, that can be traced back to Lebesgue. It was introduced, in the framework of Sobolev spaces, by Manfredi, in connection with the analysis of the regularity of maps of finite distortion appearing in the theory of nonlinear elasticity. Diverse authors, including Iwaniecz, Kauhanen, Koskela, Maly, Onninen, Zhong, thoroughly investigated continuity properties of monotone functions in the more general setting of Orlicz–Sobolev spaces, in view of the analysis of continuity, openness and discreteness properties of maps under minimal integrability assumptions on their distortion. The present paper complements and augments the available Orlicz–Sobolev theory of weakly monotone functions. In particular, a variant is proposed in a customary condition ensuring the continuity of functions from this class, which avoids a technical additional assumption, and applies in certain situations when the latter is not fulfilled. The continuity outside sets of zero Orlicz capacity, and outside sets of (generalized) zero Hausdorff measure are also established when everywhere continuity fails.

中文翻译：

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弱单调 Orlicz-Sobolev 函数的连续性

摘要 弱单调函数的概念扩展了单调函数的经典定义，可以追溯到勒贝格。它由 Manfredi 在 Sobolev 空间的框架中引入，结合非线性弹性理论中出现的有限变形图的规律性分析。不同的作者，包括 Iwaniecz、Kauhanen、Koskela、Maly、Onninen、Zhong，从分析极小条件下地图的连续性、开放性和离散性的角度出发，深入研究了 Orlicz-Sobolev 空间更一般设置中单调函数的连续性性质。对其失真的可积性假设。本论文补充和增强了现有的弱单调函数 Orlicz-Sobolev 理论。特别是，在习惯条件下提出了一个变体，以确保此类功能的连续性，这避免了技术上的额外假设，并适用于不满足后者的某些情况。零 Orlicz 容量集外的连续性和（广义）零 Hausdorff 测度的外集也在处处连续性失败时建立。