check whether a set is a positively invariant set

Given a continuous function $V(x(t))$. Suppose $V(x(t)) = 0$ and $\frac{dV}{dt} = 0$ when $x(t) \in A$ with A a compact set, and $V(x(t)) > 0$ and $\frac{dV}{dt} < 0$ when $x(t) \in \mathbb{R}\setminus A$. Now assume $x(0) \in A$, will $x(t)$ stay in $A$ for all $t \geq 0$?

My idea is to use the sufficient condition for local minimum/maximum to prove that $x(t)$ will not leave $A$, i.e., $A$ is a positively invariant set. However such condition requires the existence of $x(t)$ in the neighborhood of $0$, i.e., $N(0,\delta)$.