Vanishing derivatives and nilpotency (Q1322645)

Let \(\mathcal V\) be the algebra of all real sequences \((x)= (x_ 1,x_ 2,\dots)\) with finite variation \(\sum| x_{n+ 1}- x_ n|\), \({\mathcal A}\subset {\mathcal V}\) be the ideal of all absolutely summable sequences. The author studies the nonstandard extension \(\mathbb{R}^*= {\mathcal V}/{\mathcal A}\supset \mathbb{R}\) of the real line, especially mappings \(f^*: \mathbb{R}^*\to \mathbb{R}^*\) naturally induced by smooth functions \(f: \mathbb{R}\to \mathbb{R}\). To state a typical result, denote by \([x]\in \mathbb{R}^*\) a class of sequences \((x)\in {\mathcal V}\), and let \(\mathbb{I}\subset \mathbb{R}^*\) be the ideal of all \([x]\) such that \(\lim(x)= 0\). Then \(f^{(k)}(0)\equiv 0\) for all \(k\) if and only if \(f^*(x)= 0\) for every nilpotent element \([x]\in \mathbb{I}\).