Differential properties of absolutely continuous functions (Q1100574)

Consider an absolutely continuous function y in an interval \([0,\delta)\) and suppose that there exist numbers \(a_ 0,a_ 1,...,a_ n,\quad n\geq 1,\) and a function \(\epsilon (x),\;x\in [0,\delta),\) such that \(y(x)=a_ 0+\sum^{n}_{i=1}a_ ix^ i+\epsilon (x)x^ n,\) \(x\in [0,\delta)\), \(\epsilon (x)\to 0\;if\;x\to 0.\) Define \(\epsilon_ 1(x)\) by the formula \(y(x)=\sum^{n}_{i=1}i a_ ix^{i-1}+\epsilon_ 1(x)x^{n-1}\) for every \(x\in (0,\delta)\) at which y(x) exists. Some relations between \(\epsilon_ 1\) and \(\epsilon\) are studied. It is proved e.g. that for any \(\alpha\in (0,1)\) the set \(\{x\in (0,\delta);\;\epsilon_ 1(x)\leq \epsilon_+(x)\}\) has positive Lebesgue measure. Here \(\epsilon_+(x)=\epsilon (x)\) if \(\epsilon(x)\geq 0\), \(\epsilon_+(x)=0\) if \(\epsilon(x)<0\).