A smooth map which has full rank at a point is locally a submersion/immersion. Let $F:M\rightarrow N$
  be a smooth map between manifolds and let $p\in M$
 .


*

*If $dF_{p}$
is surjective, then there exists a neighbourhood $U$
of p
such that $F|_{U}$
is a submersion. 

*If $dF_{p}$
is injective, then there exists a neighbourhood $U$
of $p$
such that $F|_{U}$
is an immersion. 
I saw this theorem in Lee's Introduction to Smooth Manifolds. This is his proof:

What I don't understand is the part where says "by continuity". Is he talking about a function that is continuous? If so, what function is it?

In coordinates, $\Bbb d F$ is represented by a matrix. Since $\Bbb d F _p$ has full rank, there exists a minor in it of maximal dimension that is non-zero at $p$. The determinant of this minor is a continuous (in fact polynomial, being a sum of products of entries in the matrix) function of the coordinates, so if it's non-zero at $p$, by continuity it will be non-zero on some small neighbourhood of $p$, so the rank will be maximal on this neighbourhood.
