= Polynomial multiples of a Gaussian flat function
{title2=$p(1/x)e^{-1/x^2}$}
Extend this expression by zero at zero. <Gaussian decay dominates every power>, so the resulting function and its derivative difference quotients vanish to every order at zero. Away from zero, differentiation replaces $p(t)$ by $2t^3p(t)-t^2p'(t)$, another <polynomial>. Induction therefore gives a <smooth function> flat at zero. If $p$ is nonzero, its <Taylor series> is zero while the function is nonzero at every sufficiently small nonzero argument, so it is not <real analytic> there.
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