I'm looking for a proof or reference proving this theorem which was written in my ODE class:
Let $D \subseteq \mathbb{R}^d$ be open and $f:D \to \mathbb{R}^d$ continuous such that each IVP has a unique solution.
Under these conditions one can define the orbit from an initial point $x_0$ of an autonomous equation $x' = f(x)$:
The orbit passing through $x_0$ is the set $\Gamma(x_0) = \{ G(t;0,x_0). t \in ]\alpha(x_0),\omega(x_0)[\}$
One can also define the notion of invariant set for $x' = f(x)$:
$B$ is an invariant set if whenever $x_0 \in B$ then $x(t;x_0) \in O$ for all $t \ge 0$ where $x$ is the unique solution for this $x_0$.
This is the theorem I need to prove:
Let $B \subseteq D$ such that $Fr(B)$ is the union of the orbits of $x' = f(x)$.
Then $B$ is an invariant set for $x' = f(x)$
My approach
Given $x_0 \in int(B)$, if $\Gamma(x_0) \subseteq B$ we finish. Otherwise, by the continuity of $\varphi$, $\exists \tau \in ]\alpha,\omega[. \varphi(\tau) \in Fr(B)$. Then the IVP $\begin{cases}x' = f(x) \\ x(\tau) = \varphi(\tau)\end{cases}$ has two solutions: $\varphi$ and $G(\cdot;\tau,\varphi(\tau))$ which lives in the frontier. By the uniqueness of solution, $\varphi = G(\cdot;\tau,\varphi(\tau)) = \Gamma(x_1)$ for some $x_1$. Therefore, $\Gamma(x_0) = \Gamma(x_1)$ and we have that $x_0 \in Fr(B)$. This contradicts $x_0 \in int(B)$.
However if $x_0 \in Fr(B) \cap B$ is not so clear.