Just for fun, I decided I'd try to prove that if a function is continuous, then . I haven't cheated by looking for a proof on the web.
Anyway, my idea is to make use of the Mean Value Theorem, namely for some . It therefore follows that
Now, in order to prove , we need to show that .
Now by the Mean Value Theorem we have
So we can let .
Proof: Let and , where . Then
Q. E. D.
My only worry is the application of the Mean Value Theorem assumes evaluating derivatives, which assumes evaluating limits. Would we have already needed to prove the result that if a function is continuous at a point then the limit of that function is equal to the function value at that point in order to make use of the Mean Value Theorem?