(I should preface this with the disclaimer that this is a slightly elaborated version of a question that I posted onto math se recently to which I received no responses, and have since deleted the question.)

I figured that this question would be among the standard topics covered in texts on dynamical systems, but surprisingly I can find very little on it and am hoping someone can point me in the right direction.

Let $x(t)$ be the trajectory of some dynamical system, say a solution to $\dot{x} = f(x)$.

Lyapunov theory lets one (among other things) reach various conclusions about the behavior of trajectories started near equilibrium points.

Instead, suppose we define a set by $G = \{x | g(x) \geq 0\}$ and we want to know when this is forward invariant. We know that (for instance) $\frac{d}{dt} g(x(t)) \geq 0$ when $g(x(t)) = 0$. I would be interested in any variant of this using instead strict inequalities/ or equalities for any of the conditions.

I know that certain forms of this statement are easy to show, but I can also see that others may be quite subtle. For instance, when relaxing strict inequality constraints, or the situation that $g$ does not have the same left and right hand derivatives. Any reference to a theorem name or book would be helpful.