Graduate Smoothness of multivariable function

[JYM]

Let $h$ be a bump function that is $0$ outside $B_\epsilon^m(0)$ and posetive on its interior.

Let $f$ be smooth function on $B_{2\epsilon}^m(0)$.

Define $f^*(x)=h(x)f(x)$ if $x\in B_{2\epsilon}^m(0)$ and $=0$ if $x\in \mathbb{R^m}-B_\epsilon^m(0)$.

I want to show that $f^*$ is smooth on $\mathbb{R^m}$.

I proceed as follows.

Clearly $f^*$ is smooth on $x\in B_{2\epsilon}^m(0)$ as it is equal to $hf$ there, and also smooth on $\mathbb{R^m}-\overline{B_\epsilon^m(0)}$ as it is zero there.

Moreover, on the open intersection $B_{2\epsilon}^m(0)-\overline{B_\epsilon^m(0)}$ both definitions agree.

Thus $f^*$ is smooth on $\mathbb{R^m}$. Is my reasoning correct? Thanks.

 

[mathwonk]

sounds exactly right, but hard to read.

 

[JYM]

Thanks.