Flat function/exp -1 by x/Taylor series/Example

We consider the function

$f\colon \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto f(x),$

given by

${}f(x):={\begin{cases}0,\,{\text{ if }}x\leq 0\,,\\e^{-{\frac {1}{x}}},\,{\text{ if }}x>0\,.\end{cases}}\,$

We claim that this function is infinitely often differentiable, which is only in ${}0$ not directly clear. We first show, by induction, that all derivatives of ${}e^{-{\frac {1}{x}}}$ have the form ${}p{\left({\frac {1}{x}}\right)}e^{-{\frac {1}{x}}}$ with certain polynomials ${}p\in \mathbb {R} [Z]$, and that therefore the limit for ${}x\rightarrow 0,\,x>0$ equals ${}0$ (see exercise and exercise). Therefore, the limit exists for all derivatives and is ${}0$. So all derivatives in ${}0$ have value ${}0$, and therefore the Taylor series in ${}0$ is just the zero series. However, the Function ${}f$ is in no neighborhood of ${}0$ the zero function, since ${}e^{-{\frac {1}{x}}}>0$.