A few of my friends had enrolled for an Analysis-I course and had recently written their final exams. One of the questions they were asked to prove was(I am writing it from memory, so forgive me if its meaningless):

If f is uniformly continuous over , then prove that there exists constants a and b such that

I had not enrolled for the course, but I know the definition of uniform continuity and a couple of simple theorems about it. This problem looked too harsh on uniformly continuous functions and I thought it was a misprint ......But it was not

Any ideas on how to solve it?

Do I need anything other than the definition? if yes then please tell me the proof. Else give me a hint.

Thank you,

Srikanth