Functions that are Lipschitz in the small

Let X and Y be metric spaces. A function f : X → Y is said to be Lipschitz in the small if there are r > 0 and K < ∞ so that d ( f ( u ) , f ( v ) ) ≤ K d ( u , v ) for any u , v ∈ X with d ( u , v ) ≤ r . We find necessary and sufficient conditions on a subset A of X such that f | A is Lipschitz for every function f that is Lipschitz in the small on X . We also find necessary and sufficient conditions on X for * L S X to be linearly order isomorphic to Lip ( Y ) for some metric space Y .