 Solution

ID: 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/solution

For

x \neq = 0

, induction gives

f^{(n)} (x) = P_{n} (1/ x) e^{- 1/ x^{2}}, P_{0} (t) = 1, P_{n + 1} (t) = 2 t^{3} P_{n} (t) - t^{2} P_{n}^{'} (t),

where each

P_{n}

is a polynomial. This recurrence follows from the product rule and

d (1/ x) / d x = - 1/ x^{2}

x \to 0 lim f^{(n)} (x) = 0 for every integer n \geq 0.

These limits also describe the derivatives of the extension at zero. If

f^{(n)} (0) = 0

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.

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