Smooth invariance of the manifold boundary

If $n=0$, [L1] makes the claim immediate. Assume $n\ge 1$. At any $p\in U$, choose Euclidean extensions of $f$ and $f^{-1}$ near $p$ and $f(p)$. Since both half-space composites are the identity, [L2] gives $D(f^{-1}{)}_{f(p)}D{f}_p=I$ and $D{f}_{p}D(f^{-1}{)}_{f(p)}=I$.

For $n\ge 1$, if a face point $p$ mapped to the relative interior, then on a Euclidean neighbourhood of $f(p)$ contained in $V$, the last coordinate of an extension of $f^{-1}$ would be nonnegative and would vanish at the interior point $f(p)$. Its differential there would therefore be zero, contradicting the invertibility from step 1.1. Applying the same argument to $f^{-1}$ proves the converse. Hence face and relative-interior points are preserved in every dimension, making the manifold boundary and interior intrinsic.