Help with understanding why limit implies uniqueness

[MathStudent999]

TL;DR Summary

I was hoping to find more clarification on uniqueness results for autonomous ODEs

I'm studying ODEs and have understood most of the results of the first chapter of my ODE book, this is still bothers me. Suppose
$$\begin{cases}
f \in \mathcal{C}(\mathbb{R}) \\
\dot{x} = f(x) \\
x(0) = 0 \\
f(0) = 0 \\
\end{cases}.
$$

Then,
$$
\lim_{\varepsilon \searrow 0}\left|\int^\varepsilon_0 {dy \over f(y)}\right| = \infty
$$
implies solutions are unique. Since
$$
\lim_{\varepsilon \searrow 0}\left|\int^\varepsilon_0 {dy \over f(y)}\right|< \infty
$$
allows us to invert to get a solution(more clarification on this) other than 0. So, am I seeing it right that this is just a contrapostive to get uniqueness.

 

[andrewkirk]

If the book presents it as you describe, then your concerns are well-founded, as it makes the logical fallacy of "denying the antecedent" (I can't post a link to the wiki page on this locked-down computer), which is erroneously concluding $\neg P\to \neg Q$ from $P\to Q$.

In this case, $P$ is the inequality in the OP and $Q$ is the claim "there exists more than one solution".

From $P\to Q$ one can conclude $\neg Q\to\neg P$ but one cannot conclude $\neg P\to\neg Q$. eg consider where $P$ is "Beryl was born in Bulgaria" and $Q$ is "Beryl was born in Europe".
Beryl may have been born in Poland.

I expect there are other arguments that can justify the book's conclusion, but the author didn't notice the logical fallacy of the above, and hence omitted to state them.