Prove the following: Let be real numbers, let and be functions which are differentiable on . Suppose that , that is nonzero on , and exists and equals . Then for all , and exists and equals .

Suppose that for some . Since , then by Rolle's theorem we can find a such that . But this contradicts the fact that is nonzero on . Hence for all .

Now to show that exists and equals , we can show that for any sequence which converges to . How would I show this? I am guessing that I have to create a new function, and somehow manipulate it so that . And this is equivalent to showing that .