Lipschitz continuity proof given boundedness of a derivate (not a derivative)

Hey guys. Tough problem here (tough for me anyway...)

Quote:

Suppose the derivate

of a function

on

is bounded, where

.

Show that

is Lipschitz continuous.

I can show that is of bounded variation, and therefore is differentiable almost everywhere. I don't know if that's a promising approach, however, nor even if it is, how to finish the proof.

One other possible avenue is this: The proof that a function is Lipschitz if its derivative is bounded uses the mean value theorem. Is there maybe some variation of the mean value theorem for which I could use in this case?

Any help would be much appreciated!