The theorem is stated as follows:
Assume that is continuous and that and have opposite signs. Then there is a point such that .
The beginning of the proof is as follows:
Consider the case where . Let and set . Observe first that since is continuous and , then .
I don't see why follows from the fact that f is continuous and , but I would really like to understand this, especially since the author seems to think it needs no further explanation. I did try looking at other proofs for the IMT, on this site and other sites, but none of them phrased the proof in this way and they didn't really further my understanding.