This study considers the uniqueness problem of the preference relation corresponding to a demand function, which is called the “recoverability problem”. We show that if a demand function has sufficiently wide range and is income-Lipschitzian, then there exists a unique corresponding upper semi-continuous preference relation. Moreover, we explicitly construct a utility function that represents this preference relation. Compared with related research, a feature of our result is that it ensures not only the uniqueness, but also the existence of the corresponding upper semi-continuous preference relation. Further, we introduce two axioms related to demand functions, and show that these axioms are equivalent to the continuity of our preference relation in the interior of the consumption set. In addition to these results, we present three examples that explain why our requirements (including the upper semi-continuity of preference relations and the wide range requirement and income-Lipschitzian property of demand functions) are necessary, and a further two examples in which there is no continuous preference relation corresponding to the given demand function.

该研究考虑了与需求函数相对应的偏好关系的唯一性问题，称为“可恢复性问题”。我们表明，如果需求函数具有足够宽的范围，并且是收入-李普希兹函数，那么就存在一个唯一的对应的上半连续偏好关系。此外，我们显式构造了一个表示此偏好关系的效用函数。与相关研究相比，我们的结果的一个特点是它不仅确保了唯一性，而且确保了相应的上半连续偏好关系的存在。此外，我们介绍了两个与需求函数有关的公理，并表明这些公理等同于我们在消费集内部的偏好关系的连续性。除了这些结果，