Consider the following statement:


*If $f:\mathbb{R} \rightarrow \mathbb{R}$ is a continuous function, for the autonomous equation $$x' = f (x)$$ 
the "Peano phenomenon" can arise only at those values of $\bar x$ for which $f(\bar x) = 0$.*


(The "Peano phenomenon" = Cauchy problems associated to the above equation can admit more than one solution.)


Can someone provide me with a (proof or a) reference for a proof ? According to [this](https://www.emis.de/journals/PM/57f4/pm57f402.pdf) paper, this statement is "well known", so a proof of it should be found in textbooks, but I couldn't find any. A paper that contains the proof is mentioned in the linked paper above, but I don't have access to that paper.



The "Peano phenomenon" for differential equations
I'm not sure this is "well known" (it wasn't to me). We can do this with a [Lyapunov function](https://en.wikipedia.org/wiki/Lyapunov_function) $V(x,y)=g(y)-g(x)$, with $g(x)=\int\_a^x (1/f(u))\, du$. This is well defined near an $a$ with $f(a)\not= 0$.


Now if $x(t),y(t)$ both solve $x'=f(x)$, $x(0)=a$, then $W(t)= V(x(t),y(t))$ satisfies $W(0)=0$,
$$
W'(t)= g'(y)f(y)-g'(x)f(x)= 0 ,
$$
by our choice of $g$. It follows that $W\equiv 0$, and this gives uniqueness because $g$ is strictly monotone.


