Differentiation of $n$-convex functions
Volume 209 / 2010
Fundamenta Mathematicae 209 (2010), 9-25
MSC: Primary 26A24; Secondary 26A51, 26A12, 26C99.
DOI: 10.4064/fm209-1-2
Abstract
The main result of this paper is that if $f$ is $n$-convex on a measurable subset $E$ of $\mathbb R$, then $f$ is $n-2$ times differentiable, $n-2$ times Peano differentiable and the corresponding derivatives are equal, and $f^{(n-1)}=f_{(n-1)}$ except on a countable set. Moreover $f_{(n-1)}$ is approximately differentiable with approximate derivative equal to the $n$th approximate Peano derivative of $f$ almost everywhere.
