Non-analytic smooth function

Consider the real function

$f(x)=\left\{{\begin{matrix}e^{-1/x^{2}}&{\mbox{if}}\ x\neq 0\\\\0&{\mbox{if}}\ x=0\end{matrix}}\right\}.$

One can show that f has derivatives of all orders at every point including 0. To show this at x = 0, use L'Hopital's rule, mathematical induction, and some simple substitutions. But in proving this, one will find that f⁽ⁿ⁾(0) = 0 for all n. Therefore, the Taylor series of f is zero at every point! Consequently f is not analytic at 0. This pathology cannot occur with functions of a complex variable rather than of a real variable.