Let $$\dot{x}(t)=f(x(t))$$ be a dynamical system where $x\in \mathbb{R}^{n}$, assuming that the safe set $\mathcal{C}$ is the Level Set of a smooth function $h:\mathbb{R}^{n}\to \mathbb{R}$, i.e. $$\mathcal{C}:=\{ x\in \mathbb{R}^{n}:h(x)\ge 0 \}$$ and that $$\frac{ \partial h }{ \partial x } (x)\not=0,\,\forall x\in \mathbb{R}^{n}:h(x)=0.$$Then, Nagumo’s Theorem gives necessary and sufficient conditions for set invariance based upon the derivative of $h$ and the boundary of $\mathcal{C}$: $$\mathcal{C}\text{ invariant}\iff \dot{h}(x)\ge0\,,\forall x\in\partial \mathcal{C}$$