I'm learning multivariate analysis from Analysis on Manifolds written by Munkres. In solving the problems, I have encountered this problem:
In (a), since the support of $\phi$ is in $U$, if $\mathbf{x}\notin K:=\operatorname{supp}\phi$, $\phi(x)=0$. Hence the definition makes sense. But how can I show that this $h$ is of class $C^r$? On $K^c$, $h$ is identically zero, which implies that $h$ is $C^r$ on $K^c$. But on $K$, I don't know how to show that $h$ is of $C^r$.
