Exercise 10.6

Following the hint, recall that Exercise 8.1 defined an infinitely differentiable function on $1$ such that $f(x)=0$ for $x\le 0$ and $0\le f(x)<1$ for all $x$. Let $a<b$. Then the function $g_{a,b}(x)=f(x-a)f(b-x)$ is also infinitely differentiable, equals 0 for $x\le a$ and $x\ge b$, and $0\le g_{a,b}(x)<1$ for all $x$. Since it has compact support, we can define a function

which is infinitely differentiable, equals 1 for $x\le a$, equals 0 for $x\ge b$, and $0\le h_{a,b}(x)\le 1$ for all $x$.

Now let $\check{x}\in n$, and let $B(\check{x})$ and $W(\check{x})$ be open balls centered at $\check{x}$ with radii $a<b$, respectively. Define the function $r(\check{y})$ for $\check{y}\in n$ which is the distance between $\check{x}$ and $\check{y}$, that is,

which is infinitely differentiable for $\check{y}\ne \check{x}$. Then the function $\phi =h_{a,b}\circ r$ is infinitely differentiable on $n$, equals 1 on $\overline{B(\stackrel{ˇ}{x})}$, equals 0 on $W(\check{x})$, and $0\le \phi (\check{y})\le 1$ for all $\check{y}$. We can use these functions in the proof of Theorem 10.8 to get infinitely differentiable functions ${\psi }_i$.