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.
By the first step the rank is locally constant. Since locally constant functions on connected spaces are constant, the rank is constant.