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

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

Quote:

Suppose the derivate $\displaystyle D^+$ of a function $\displaystyle f$ on $\displaystyle [a,b]\subseteq\overline{\mathbb{R}}$ is bounded, where

$\displaystyle D^+=\limsup_{h\to 0^+}\frac{f(x+h)-f(x)}{h}$.

Show that $\displaystyle f$ is Lipschitz continuous.

I can show that $\displaystyle f$ 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 $\displaystyle D^+$ which I could use in this case?

Any help would be much appreciated!