Consider the differential equation $\dot{x}=f(x)$, where $f: \mathbb{R}^2 \to \mathbb{R}^2$ is smooth. Given a set $A \subset \mathbb{R}^2$, are there some results saying that whenever $x(0) \in A$, there exists $T>0$, such that $x(T) \notin A$ (so every trajectory starting from $A$ will always leave $A$ at some time instant)?

Note: I do not specify what the set $A$ looks like (e.g., compact, convex, etc) because I want to know all the possibilities. This means, your answer can, of course, require that $A$ is convex or whatever you like, as long as the answer is correct. And I also do not specify the behavior of the trajectory after $T$ because it is not important in this question. The question itself is clear enough.