Let $ f(x)= \begin{cases} \exp\left(\frac{-1}{1-|x|^2}\right), &\text{ if } |x| < 1, \\ 0, &\text{ if } |x|\geq 1. \end{cases} $ Prove that $f$ is infinitely differentiable everywhere. ($x$ belongs to $\mathbb{R}^n$ for fixed $n$.)
Well, this is obvious for $|x|>1$ and easy enough for the first derivative at $|x|=1$, but I can't seem to use the definition of the Gateaux derivative to show it for $|x|<1$. Any advice would be appreciated.
(This is not homework.)