Suppose $\dot{x}=f(x)$ is a dynamical system, with $x$ in $R^n$, and $f:R^n \to R^n$ sufficiently smooth (for example, Lipschitz-continuous).

Assume that $x_e$ is an unstable equilibrium point of the system. Even if $x_e$ is unstable, the union of all trajectories having $x_e$ as a limit point could be larger than just the singleton {$x_e$}. But does this set always have zero measure?

This might be a basic result, but any pointer would be appreciated.