The closed unit ball is a smooth manifold with boundary

Let $m$ be any natural number, $D=\{x\in\mathbb R^{m+1}:\|x\|\leq1\}$, $H=\{y\in\mathbb R^{m+1}:y_0\geq0\}$, and $s=2e_0$. For each unit vector $p$, let $L_p:(-p)^\perp\to\mathbb R^m$ be the coordinate isometry from the orthonormal basis selected by the construction, put $S_p(u)=L_p\bigl(\frac{2}{1+\langle p,u\rangle}(u-\langle p,u\rangle p)\bigr)$ and $U_p(t)=\frac{4L_p^{-1}(t)+(4-\|t\|^2)p}{\|t\|^2+4}$, and define $N_p(0)=p$, $N_p(x)=x/\|x\|$ for $x\ne0$. In these total formulas a reciprocal of zero is zero. The interior chart has forward map $x\mapsto x+s$, inverse $y\mapsto\min\{1,\|y-s\|^{-1}\}(y-s)$, source $\{x\in D:\|x\|<1\}$, and target $\{y\in H:\|y-s\|<1\}$. The chart indexed by $p$ has forward map $x\mapsto(1-\|x\|,S_p(N_p(x)))$, inverse $(a,t)\mapsto\max\{0,\min\{1,1-a\}\}U_p(t)$, source $\{x\in D:x\ne0,\ x\ne\|x\|(-p)\}$, and target $\{(a,t)\in H:a<1\}$. Give $D$ the charted-space structure with exactly that atlas and preferred chart equal to the interior chart at $\|x\|<1$, and otherwise equal to the chart indexed by $N_{e_0}(x)$, where $e_0$ is the first standard unit vector. For every order $k$ that is a nonnegative integer, $\infty$, or $\omega$, this charted space satisfies the $C^k$ manifold condition for the half-space model $I:H\to\mathbb R^{m+1}$ whose forward map is inclusion and whose total inverse replaces coordinate zero by its maximum with zero. Specifically, every atlas transition $\psi=c'\circ c^{-1}$ has the property that $I\circ\psi\circ I^{-1}$ is $C^k$ on $I(\operatorname{source}\psi)$ and $I\circ\psi^{-1}\circ I^{-1}$ is $C^k$ on $I(\operatorname{target}\psi)$. The orders $k=0$, $\infty$, and $\omega$ include topological, smooth, and analytic regularity, respectively; the case $m=0$ is included, and empty chart overlaps give vacuous regularity conditions.