I have the following exercise to fullfill:
Given the system of differential equations $x'=f(x)=-\nabla{g}$ , where $x(t)\in\Bbb{R^3}$ and $g$ is $C^1$ and $f(0)=0$ and $0$ is a total maximum for $g$, decide about the stability of the point $0$.
My attempt: I consider the function $V=g(0)-g(x)$. Now, If the $0$ is an isolated point of maximum of $g$ and isolated equilibrium point of $f$ , we conclude that $V$ is a Lyapounov function with $\nabla{V}\cdot{f}$ $=\nabla{g}\cdot\nabla{g}$ , which is positive for $x\neq0$. So according to the Lyapunov theorem the $0$ is unstable.
My question is if we can solve the exercise given by not using the fact that $0$ is isolated, as stated in my proof. Thanks.