I don't think so. Every function which is differentiable and has bounded derivative is uniformly continuous (Wikipedia), and it is possible to have a function that is above y = x and has a bounded derivative. The problem with is that its derivative is not bounded on all . In fact, the link above has an outline of the proof that is not uniformly continuous.

This is correct, but you are interested in the lower, not upper bound on : you want to show that it is .so we must prove that there is so for all

there is x,y in R which and