= Radial Gaussian flat function
{title2=$F(x)=e^{-1/|x|^2}\ (x\ne0),\quad F(0)=0$}
In any finite-dimensional <Euclidean space>, this extension is a <smooth function> and all its <partial derivatives> at the origin vanish. Away from zero, repeated <differentiation> gives finite sums of coordinate <polynomials> divided by powers of $|x|$, multiplied by $e^{-1/|x|^2}$. Every such term is bounded in absolute value by $C|x|^{-N}e^{-1/|x|^2}$ near zero. Since <Gaussian decay dominates every power>, this bound tends to zero faster than any prescribed positive power of $|x|$. Induction using difference quotients then proves smoothness and zero <derivatives> of every order at zero.
The origin is nevertheless a strict <global minimum>, because $F(x)>0$ for $x\ne0$. All its <Taylor series> coefficients vanish, so the function is not <real analytic> at zero. Adding a constant preserves the stationary minimum but changes the zeroth-order value. An expression written only for $x\ne0$ has no stationary point at the excluded origin until an extension is specified.
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