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Let's consider the open ball $B\left ( \omega ,r \right )\subset \mathbb{R}^{n}$. Let $f(x)$ equal $\exp\left(\frac{1}{\left | x-\omega \right |^{2}-r^{2}}\right)$ if $x\in B$, and $0$ if $x\in \mathbb{R}^{n}\setminus B$. Prove that $f$ is infinitely differentiable. The idea I had in mind was to find a general formula for the $n$th derivative, but I couldn't do it.

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The problem is on the boundary of the ball, i.e. when $\left(\sum\limits_{k=1}^n (x_k-\omega_k)^2\right)^{1/2}=r$. Consider function $g(t)=\exp\left(\frac{1}{t^2-r^2}\right)$, then $$f(x)=g\left(\left(\sum\limits_{k=1}^n (x_k-\omega_k)^2\right)^{1/2}\right).$$ Try to prove that $g$ is smooth. – Norbert Mar 10 '12 at 7:12
up vote 2 down vote accepted

In this question, it's shown when $\omega=0$ and $r=1$. We define $g(x):=f\left(r(x+\omega)\right)$, and we know that $g$ is smooth. We deduce that so is $f$, as the map $x\mapsto r(x+\omega)$ is smooth and bijective, with smooth inverse (for the composition).

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