Another almost identical method would be to assume that there exists an interval such that . But since is differentiable we can find a such that which is a contradiction.
This is called Lipschitz's continuity by the way.
If you have any questions at all feel free to ask
reversing things, i.e. you choose c and then you claim that there exists an interval [a,b] containing c with that condition! i'm not sure if this is always possible!
if there exist such then: which is possible iff because:
regarding Larrioto's question:
by mean value theorem for any real numbers there exists (between ) such that thus: