Let $B: \mathbb{R}^n\to \mathbb{C}; B(x) = \exp(\frac{1}{|x|^2-1})$ when $|x|<1$ and $B(x)=0$ when $|x|\geq 1$. Prove $B(x)$ is smooth.
Here's my idea. $B$ is radial so $B$ will be smooth on $\mathbb{R}^n \setminus \{0\}$ if $B$ is smooth in the case $n=1$. Hence, except for differentiability at the origin, we can assume $n=1$.
Now I can get this done with some tedious induction. Just wondering if there is a better way to go about this? Many thanks!