On approximate Peano derivatives
A function \(f: \mathbb{R}\to \mathbb{R}\) is said to be \(k\) times approximately Peano differentiable at \(x\in \mathbb{R}\) if there are numbers \(f_ 1(x),\dots, f_ k(x)\) and a set \(V_ x\) of density 1 at \(x\) such that \[ f(x+ t)= f(x)+ {t\over 1!} f_ 1(x)+\cdots+ {t^ k\over k!} f_ k(x)+ t^ k \varepsilon_ k(x,t), \] where \[ \lim_{x+ t\in V_ x,t\to 0} \varepsilon_ k(x, t)= 0. \] The main result of this paper reads as follows: If \(f: \mathbb{R}\to \mathbb{R}\) is \(k\) times approximately Peano differentiable at each \(x\in \mathbb{R}\), then there is a sequence of closed sets \((E_ n)_{n\in \mathbb{N}}\) so that \(\bigcup^ \infty_{n=1} E_ n= \mathbb{R}\) and \(f_{k-1}|_{E_ n}\) is differentiable and \(f_{k-1}^ \prime |_{E_ n} (x)= f_ k(x)\) for each \(n\in \mathbb{N}\). The author gives some applications of this theorem.
scientific article; zbMATH DE number 638218