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 $k$?
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 $k$?