Suppose $F:M\to N$ is a smooth map, with $M,N$ smooth manifolds and $M$ connected. I want to show that if for each $p$ in $M$ there exists smooth charts near $p$ and $F(p)$ in which the coordinate representation of $F$ is linear, then $F$ has constant rank. The first step is to show that the rank of $F$ is constant in a neighborhood of each point because $F$ is linear. The next is to show $F$ has constant rank in all $M$ by connectedness hypothesis.
How can I use the connectedness of $M$ to say that the rank is constant on all of $M$? This comes from the book of "Introduction to Smooth Manifolds" by Lee.