Let $F:\mathcal{U}\subseteq\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$, $n≤m$, $F\in\mathcal{C}^{\infty}(\mathcal{U})$. Let $Crit(F):=\lbrace p\in\mathcal{U}\mid \ dF_{p} \ is \ not \ surjective.\rbrace$. Prove that $Crit(F)$ is closed.
I thought to define a function
$$h:\mathcal{U}\rightarrow\mathbb{R}$$
$$h(p)=det(A_{p})$$
Where $A\in Mat_{2}(\mathbb{R})$ is an arbitrary minor $n\times n$ of $dF_{p}$.
$h$ is continuous and $Crit(F)=h^{-1}(0)$ so is closed. Is it right?