If $f: (a,b) -> \mathbb R$ (where $a,b$ may be $\infty$) and $f$'s derivative is bounded, then $f$ is uniformly continuous.
Why does the author specify "open interval"? Doesn't it equally apply when the interval is [0,5] or (0,5]?
If $f$'s domain is [0,5] and its derivative is bounded in this closed interval, then it seems like the above proposition must be tweaked to apply, since there is no open interval in the domain that contains the domain. Of course in this case we can just apply the Uniform Continuity Theorem ("continuous function on a compact interval implies UC"), but I am trying to better understand the conditions of this proposition. What if the interval in question was half-open like [4,5) ?
You are right, I think that the author chose open intervals so that their extremes may be ±∞. Moreover, if you take intervals with extremes then you have to introduce right and left derivatives and all of that. This is a complete waste of time since, as you know very well, a continuous function is already u.c. on closed and bounded intervals.