Let $\varphi$ be the function defined on $\mathbb{R}$ by
$$\forall t \in \mathbb{R}, \quad \varphi(t) = \begin{cases} 0 & \text{if } t \leqslant 0 \\ \mathrm{e}^{-1/t} & \text{otherwise} \end{cases}$$
Show that $\varphi$ is of class $\mathcal{C}^\infty$ on $\mathbb{R}$.
One may show that: $\forall k \in \mathbb{N}, \exists P_k \in \mathbb{R}[X], \forall t > 0, \varphi^{(k)}(t) = P_k(1/t)\mathrm{e}^{-1/t}$.