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 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.