Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2012 / Paper 1 / 19 / a / iii

For $x\ne 0$, induction giveswhere each $P_n$ is a polynomial. This recurrence follows from the product rule and $d(1/x)/dx=-1/x^2$. Since Gaussian decay dominates every power,These limits also describe the derivatives of the extension at zero. If $f^{(n)}(0)=0$, its next difference quotient is $P_n(1/h)e^{-1/h^2}/h$, which again tends to zero. Induction proves existence and continuity of every derivative, all zero there. This is a flat function, covered more generally by polynomial multiples of a Gaussian flat function.