A bounded derivative implies uniform continuity. However, there are certain instances of functions (most notably ) that are uniformly continuous and have unbounded derivatives. I attempt to strengthen the theorem below:
Theorem: An everywhere continuous function is uniformly continuous on if its derivative is bounded on and ( ).
Proof: On we have a continuous function defined on a compact set, so it is uniformly continuous and ... On and , has a bounded derivative so there exist and such that... Therefore taking proves that the entire function is uniformly continuous on .
So what's my question? My question is whether the above theorem is an if-and-only-if statement or whether the implication only goes in one direction. If not, what hypotheses can we add so that it does go both ways?