07-06-2011, 10:43 PM
You need the continuity on a compact set, but you don't need the differentiability at the endpoints, because you can find the point you want where the derivative is as you want it on the inside.
If you made it differentiable on the compact set, you'd be adding a hypothesis you don't need, that it's also differentiable at a and b.
If you made it differentiable on the compact set, you'd be adding a hypothesis you don't need, that it's also differentiable at a and b.