EDB — 1F1

[1F1] Difficulty:*.Note:Hadamard’s lemma.

Let \(f:ℝ→ℝ\) be a function of class \(C^∞\), and such that \(f(0)=0\). Define, for \(x≠ 0\), \(g(x){\stackrel{.}{=}}f(x)/x\). Show that \(g\) can be prolonged, assigning an appropriate value to \(g(0)\), and that the prolonged function is \(C^∞\). What is the relationship between \(g^{(n)}(0)\) and \(f^{(n+1)}(0)\)?