On a compact set, it is easily shown that all continuous functions are uniformly continuous. If is a bounded noncompact subset of , then there exists that is continuous but not uniformly continuous. As a simple example, consider defined by . By choosing points close to 0, we can always make for any single choice of , for a given .
On a compact set, it is easily shown that all continuous functions are uniformly continuous. If is a bounded noncompact subset of , then there exists that is continuous but not uniformly continuous. As a simple example, consider defined by . By choosing points close to 0, we can always make for any single choice of , for a given .