1. The problem statement, all variables and given/known data This should be easy but I'm stomped. Let K be a compact set in a normed linear space X and let f:X-->X be locally Lipschitz continuous on X. Show that there is an open set U containing K on which f is Lipschitz continuous. 2. Relevant equations locally Lipschitz means that for every x in X, there is a nbdh around x on which f is Lipschitz. 3. The attempt at a solution The obvious thing to do it seems it take an open cover of K by sets on which f is Lipschitz continuous and extract a finite subcover. But then what?!?