An additivity theorem for uniformly continuous functions

We consider metric spaces X with the nice property that any continuous function f:X → R which is uniformly continuous on each set of a finite cover of X by closed sets, is itself uniformly continuous. We characterize the spaces with this property within the ample class of all locally connected metric spaces. It turns out that they coincide with the uniformly locally connected spaces, so they include, for instance, all topological vector spaces. On the other hand, in the class of all totally disconnected spaces, these spaces coincide with the UC spaces.