Let $W$ denote the inverse of the bijection $f|_{[-1,+\infty[}$, where $f(x) = xe^x$. That is, for every real $x \geqslant -\mathrm{e}^{-1}$, $W(x)$ is the unique solution of $f(t) = x$ with $t \in [-1,+\infty[$. Justify that $W$ is continuous on $\left[ - \mathrm { e } ^ { - 1 } , + \infty \left[ \right. \right.$ and is of class $\mathcal { C } ^ { \infty }$ on $] - \mathrm { e } ^ { - 1 } , + \infty [$.