First of all, this conclusion is correct, which is called Cantor theorem in mathematical analysis. When I studied fractions, I proved this conclusion with the interval set theorem, which itself was discussed in a closed interval, and the conclusion of the interval set theorem could not be used in an open interval or a semi-open and semi-closed interval. Of course, there are other ways to prove the conclusion that continuous functions on closed intervals are uniformly continuous.
The continuous function on the interval (a, b) or interval (a, b) is not necessarily uniformly continuous. Example 2 in your book has given a counterexample.
I don't know how to prove Theorem 4 in your book. If you think that the closed interval in Theorem 4 can become an open interval, then the conclusion that "continuous functions are uniformly continuous on the open interval" is also correct (of course, this is wrong). If necessary, you can send the proof process of Theorem 4 in the book again, and I will find out the reason why I can't change from closed interval to open interval and explain it to you.
If you are not clear, you can ask in detail again, hoping to help you ~