Suppose that a function is differentiable . Prove that takes on every value between and .
If the above question is a misprint and wants to say "prove that takes on every value between and ", then I have no problem using the intermediate value theorem here.
If, on the other hand, it is not a misprint, then it seems to me that I can't use the Intermediate value theorem, as I can't see how I am authorised to assume that is continuous on .
Or perhaps there is another way to look at the problem?