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Can anyone explicitly verify that the function $ f(x) = \left\{ \begin{array}{lr} 0 & : x = 0 \\ e^{-1/x^{2}} & : x \neq 0 \end{array} \right. $

is infinitely differentiable on all of $\mathbb{R}$ and that $f^{(k)}(0) = 0$ for every $k$?

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    I'm sure lots of people _can_ ...2012-03-13
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    This is related http://math.stackexchange.com/questions/47948/showing-that-a-function-is-of-class-c-infty/47978#47978. You already know that that function is smooth away from zero. Now you just have to prove inductively that you have the same property in zero. The consequence of the mean value theorem showed in the link is enough to do that.2012-03-13

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