Suppose we have an autonomous differential equation $x'=f(x)$ and a connected set $S$ of fixed points (this is, $f(s) = 0$ for all $s \in S$).
Let $X$ be a set such that $\partial X = S$. Is it true that $X$ is invariant (this is, every solution to the ODE with a point in X stays in X)?
I would say it is true, because to leave $X$ any orbit should have at least a point in common with $S$, which is impossible because of the uniqueness of solutions. But I don't think I have the tools to prove it formally.
I don't really need a proof, just a comment on the veracity of this claim or a reference of related claims.