Exercise 1, § 3

The function $\varphi_0$ on $\mathbf{R}$ defined by $\varphi_0(x) = 0$ if $x\leqslant 0$ and $\varphi_0(x) =e^{-1/x}$ if $x >0$ is of class $C^{\infty}$ and verifies $\varphi^{(k)}_0(0) = 0$ for every integer $k\in \mathbf{N}$.