A function is uniformly continous on if for every there is so that if . Note, if is uniformly continous on then certainly is continous on .

An easy way to prove uniform continuity here is to note that on is a continous function. Continous functions on compact sets are uniformly continous on the set (this is a theorem). It follows that is uniformly continous on . But if it is uniformly continous on then it is surly uniformly continous on for it is a subset.