We prove that for continuous Lorentz-Finsler spaces timelike completeness implies inextendibility. Furthermore, we prove that under suitable locally Lipschitz conditions on the Finsler fundamental function the continuous causal curves that are locally length maximizing (geodesics) have definite causal character, either lightlike almost everywhere or timelike almost everywhere. These results generalize previous theorems by Galloway, Ling and Sbierski, and by Graf and Ling.

中文翻译：

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Lorentz-Finsler 空间的一些正则性结果

我们证明，对于连续的 Lorentz-Finsler 空间，时间完整性意味着不可扩展性。此外，我们证明了在 Finsler 基本函数上合适的局部 Lipschitz 条件下，局部长度最大化（测地线）的连续因果曲线具有明确的因果特征，几乎处处像光一样或几乎处处像时间。这些结果概括了 Galloway、Ling 和 Sbierski 以及 Graf 和 Ling 先前的定理。