Suppose $F:M\to N$, $M,N$ smooth manifolds and $M$ connected. I proved that for each $p$ in $M$ there exists smooth charts near $p$ and $F(p)$ in which the coordinate rapresentation of $F$ is linear. This is sufficient to prove that the rank of $F$ is constant in a neighborhood of each point. How can I use the connectedness of $M$ to say that the rank is constant on all of $M$? This is a property on the book of "Introduction to Smooth Manifolds" by Lee.
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