Set of critical points of a smooth map between two manifolds is closed

Question

Suppose, $f:M\to N$ is a smooth map whre $M$ and $N$ are any two smooth manifolds.Then how can I show that, the set of critical points of $f$ is closed in $M$.

I am not getting any hint to go forward.

Answer 1

The question is local, you can assume that $f:U\subset\mathbb{R}^n\rightarrow V\subset\mathbb{R}^m$, $x$ is a critical point if the determinant of all the minors of $df_x\in L(\mathbb{R}^n,\mathbb{R}^m)$ of rank $n$ if $n\leq m$, or rank $n-m$ if $n>m$ are zero. The map defined by $h_M(x)=det(M(x))$ where $M(x)$ is a minor is continuous.