Locally Lipschitz functions, cofinal completeness, and UC spaces

Let 〈X,d〉 be a metric space. We find necessary and sufficient conditions on the space for the locally Lipschitz functions to coincide with each of two more restrictive classes of locally Lipschitz functions studied by several authors: the uniformly locally Lipschitz functions and the Lipschitz in the small functions. In the first case, we get the cofinally complete spaces and in the second, the UC spaces. We address this question: to which family of subsets of 〈X,d〉 is the restriction of each function in each class actually Lipschitz? Finally, we determine exactly when the class of uniformly locally Lipschitz functions is uniformly dense in the Cauchy continuous real-valued functions, a class that naturally contains them. In fact, our theorem is valid when the target space is any Banach space. Our density theorem complements the uniform approximation results of Garrido and Jaramillo [12,13].