Infinite Peano derivatives. (English) Zbl 1012.26004

Summary: Let \(f_{(n)}\) and \(\underline f{}_{(n)}\) denote the \(n\)th Peano derivative and the \(n\)th lower Peano derivative of the function \(f: [a,b]\to\mathbb{R}\). We investigate the validity of the following statements.
\((M_n)\). If the set \(H= \{x\in [a,b]:\underline f{}_{(n)}(x)> 0\}\) is of positive outer measure, then \(f\) is \(n\)-convex on a subset of \(H\) having positive outer measure.
\((Z_n)\). The set \(E_n(f)= \{x\in [a,b]: f_{(n)}(x)=\infty\}\) is of measure zero for every \(f: [a,b]\to \mathbb{R}\).
We prove that \((M_n)\) and \((Z_n)\) are true for \(n= 1\) and \(n=2\), but false for \(n\geq 3\). More precisely, we show that for every \(n\geq 3\) there is an \((n-1)\) times continuously differentiable function \(f\) on \([a,b]\) such that \(f_{(n)}(x)= \infty\) a.e. on \([a,b]\), and that such a function cannot be \(n\)-convex on any set of positive outer measure.
We also show that the category analogue of \((Z_n)\) is false for every \(n\). Moreover, the set \(E_n(f)\) can be residual. On the other hand, the category analogue of \((M_n)\) is true for every \(n\). More precisely, if \(\{x\in [a,b]:\underline f{}_{(n)}(x)> 0\}\) is of second category, then \(f\) is \(n\)-convex on a subinterval of \([a,b]\). As a corollary we find that \(E_n(f)\) cannot be residual and of full measure simultaneously.


26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems