A function f from a metric space $(X,d)$ to another metric space $(Y,\rho )$ is said to be Cauchy-continuous if for every Cauchy sequence $(x_n)$ in $(X,d)$, $(f(x_n))$ is Cauchy in $(Y,\rho )$. It is well known that a metric space is complete if and only if every real-valued continuous function defined on it is Cauchy-continuous. Here we consider a well-studied intermediate class of metric spaces which lies between the class of compact metric spaces and that of complete metric spaces called cofinally complete metric spaces. In this paper, we use Cauchy-continuous functions and some Lipschitz-type functions to study a more general case, that is, to study the metric spaces which have a cofinal completion. We also study a geometric functional that measures the local total boundedness of a metric space at its each point and characterize the aforesaid metric spaces using it.

度量空间中的函数f$（X，d）$ 到另一个度量空间 $（ÿ，\rho ）$ 如果对于每个柯西序列，则被称为柯西连续的 $（X_{ñ}）$ 在 $（X，d）$， $（F（X_{ñ}））$ 是柯西在吗 $（ÿ，\rho ）$。众所周知，当且仅当在其上定义的每个实值连续函数都是柯西连续的时，度量空间才是完整的。在这里，我们考虑一个经过精心研究的度量空间中间类，它介于紧凑度量空间类和称为共同最终度量空间的完整度量空间类之间。在本文中，我们使用柯西连续函数和一些Lipschitz型函数来研究更一般的情况，即研究具有共终完成的度量空间。我们还研究了一种几何函数，它测量度量空间在其每个点的局部总有界度，并使用它来表征上述度量空间。