Rank of Jacobian is lower semicontinuous

Statement

Let $M$ and $N$ be differential manifolds and $f:M\to N$ be a smooth map. Define the following map:

$rk(Df):M\to \mathbb {N} _{0}$

which sends $p\in M$ to the rank of $Df:T_{p}M\to T_{f(p)}(N)$.

The function function $rk(Df)$ is lower semicontinuous. In other words, given any point $p$, there exists an open set $U$ containing $p$ such that $rk(Df)(q)\geq rk(Df)(p)$ for all $q\in U$.

Equivalently, for any $r\in \mathbb {N} _{0}$, the set of points in $M$ for which $rk(Df)\geq r$, is an open subset.