Choose any value , we know there exists a corresponding interval such that and since was arbitrary this must work for every interval .
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
this, although an interesting claim, but is not what mean value theorem says: you choose an interval [a,b] and then you find a < c < b that satisfies the condition. what you're doing here is
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!
here's a simple counter-example: let and choose geometrically it's clear that there are no such that you can also prove it very easily:
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: