## Real Analysis Exchange

### Infinite Peano Derivatives

Miklós Laczkovich

#### Abstract

Let $f_{(n)}$ and ${\underline f}{_{(n)}}$ denote the $n^{\rm th}$ Peano derivative and the $n^{\rm th}$ 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\ge 3.$More precisely we show that for every$n\ge 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.

#### Article information

Source
Real Anal. Exchange, Volume 26, Number 2 (2000), 811-826.

Dates
First available in Project Euclid: 27 June 2008