I know that is NOT uniformly continuous, but isn't it u.c. on [0,1]. How would I prove this directly using the definiton? Unless its not uniformly continuous on [0,1] then how would I prove that using the sequential criterion?
You need to know that from any collection of open intervals that cover there is a finite subcollection which also covers .
Do you have that theorem?
Also, is the first part of your question asking why isn't uniformly continuous on ? I'll leave you to think about it, but for your benefit I'd like to remark that it's (fairly) easy to prove that every uniformly continuous function on is sublinear (i.e. for some constants ) from where it follows that is not uniformly continuous on for every .