Problem 48 says:
Consider the theorem: If f and f' are both strictly increasing real-valued functions on (0,infty), then
. The following argument is suggested as a proof of this theorem.
(1) By the Mean Value Theorem, there is a
in the interval (1,2) such that
and then a bunch more steps.
According to the answer key, this is a valid argument. I must be missing something huge here though, because I was under the impression that the mean value theorem required f(x) to be continuous on [a,b] and f'(x) to be continuous on (a,b). Here all we have is that the functions are strictly increasing. What am I doing wrong?