2 Added the requirement that the dynamical system $\dot{x} = f(x)$ have a semiflow.
Theorem. Consider a continuous map $f : {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n}$ and suppose that the autonomous dynamical system $\dot{x} = f(x)$ has a nonempty subset semiflow $\varphi : {\mathbb{R}}_{\geq{0}} \times {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n}$. Let $K \subseteq {\mathbb{R}}^{n}$. If $K$ is forward-invariant for the autonomous dynamical system $\dot{x} = f(x)$nonempty, and if $K$ is compactand , convex and forward-invariant, then $K$ contains an equilibrium of the dynamical system$\dot{x} = f(x)$, i.e. a zero of the map $f$.
Theorem. Consider a continuous map $f : {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n}$ and a nonempty subset $K \subseteq {\mathbb{R}}^{n}$. If $K$ is forward-invariant for the autonomous dynamical system $\dot{x} = f(x)$, and if $K$ is compact and convex, then $K$ contains an equilibrium of the dynamical system $\dot{x} = f(x)$, i.e. a zero of the map $f$.