If $f(x)=\chi_{(0,\infty)}\exp(-1/x)$, show that $f\in C^{\infty}$.
Show that $f(x) = 0$ if $x = 0$ $f(x) = \text{exp}(-1/x^{2})$ if $x \neq 0$ is infinitely differentiable but not analytic at $x_{0} = 0$ `