实函数的柯西-哈梅尔连续性
On the Cauchy-Hamel Continuity of Real Functions
AI总结：
本文比较多种使加性函数连续的连续性概念，确定Q-连续性的双边版本为柯西-哈梅尔连续性的最优候选，相关结果为力学平行四边形法则的奇异模型连续性问题提供了初步线索。
AI中文摘要：
我们对几种能使所有加性函数连续的“病态”连续性概念进行了比较，目的是确定其中哪种概念性质最优，值得被称为柯西-哈梅尔连续性。结论是，此前引入的Q-连续性概念的双边版本似乎是最有前景的候选者。我们的工作与力学的公理基础相关，具体涉及力的合成平行四边形法则的公理化刻画问题。达布曾指出，对该平行四边形法则的这种刻画需要连续性假设，若缺少此类连续性公理，就可能存在基于其他合成规则的“奇异”物理模型。一个引人关注的问题是，这类奇异物理模型是否仍具有某种弱的、残余的连续性概念。本文研究的概念为回答该问题提供了初步线索。
英文摘要：
We compare several "pathological" notions of continuity that make every additive function continuous. Our goal is to determine which of these notions is best behaved and deserves to be called Cauchy-Hamel continuity. The conclusion is that a bilateral version of a previously introduced notion of Q-continuity seems the most promising candidate. Our work is relevant to the axiomatic foundations of mechanics, specifically to the problem of axiomatically characterizing the parallelogram rule for the composition of forces. It was noted by Darboux that a continuity assumption is needed for such a characterization of the parallelogram rule and that, absent such a continuity axiom, "exotic" physical models based on alternative composition rules may exist. An intriguing question is whether such exotic physical models still possess some weak, residual notion of continuity. The concepts studied in this paper offer a first hint of a response to this question.
