After I reas the section about the Rolle's lemma and Mean Value Theorem, there were examples about a function that has exactly one root in a certain closed interval. The existence of a root was clear to me, and the uniquenesswas justified by Rolle's lemma. However, if the interval were not closed, but open -- and even not bounded, then the author does not explain how it could be unique, when it has been proven that the function was strictly increasing. Here is the following proposition, I would like you to confirm, as I fear that I removed too many assumptions such as the closeness/openness, continuity and differentiability... it might seem too trivial but I can not find an exercise about it in any books;
Proposition: If $f$ is a strictly monotone function in an interval $I$ in $\mathbb{R}$, then it has at most one root in the given interval.
Proof: Assume otherwise that $f$ has more than one root in $I$. Let $r_1$ and $r_2$ with $r_1<r_2$ be two consecutive roots, then $f(r_1)=f(r_2)=0$. But, this contradicts the fact that $f$ is strictly monotone. QED.
