An immersion has constant rank in some neighborhood

Question

Let $f:N\to M$ be a smooth map between two manifolds of dimension $n,m$. If $f_*$, the differential of $f$ is injective at $p$ we say that $f$ is an immersion. I want to show that $f$ has constant rank $n$ in some neighborhood of $p$. I have no idea how to get started. Any help is appreciated. Thanks!

Answer 1

HINT: If a continuously varying $m\times n$ matrix has rank $n$ at some point $p$, show that it has rank $n$ nearby. (Remember that determinant is continuous.)