More about cofinally Bourbaki quasi-complete metric spaces

We characterize cofinally Bourbaki quasi-complete metric spaces and their completions in terms of certain Lipschitz-type functions. To this end, we introduce and study a new class of functions, namely strongly uniformly locally Lipschitz functions, which lie strictly between Lipschitz functions and uniformly locally Lipschitz functions. We show that a metric space <X, d> is cofinally Bourbaki quasi-complete if and only if the class of strongly uniformly locally Lipschitz functions on <X, d> coincides with the (a priori) larger class of locally Lipschitz functions. Moreover, the completion of <X, d> is cofinally Bourbaki quasi-complete if and only if the class of strongly uniformly locally Lipschitz functions agrees with the class of Cauchy-Lipschitz functions. Finally, we provide several characterizations of cofinally Bourbaki quasi-complete metric spaces and their completions using functions that preserve certain classes of Cauchy-type sequences.