## Real Analysis Exchange

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

### Infinite Peano Derivatives

#### 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

**Permanent link to this document**

https://projecteuclid.org/euclid.rae/1214571369

**Mathematical Reviews number (MathSciNet)**

MR1844395

**Zentralblatt MATH identifier**

1012.26004

**Subjects**

Primary: 26A24: Differentiation (functions of one variable): general theory, generalized derivatives, mean-value theorems [See also 28A15]

**Keywords**

Peano derivatives $n$-convex functions

#### Citation

Laczkovich, Miklós. Infinite Peano Derivatives. Real Anal. Exchange 26 (2000), no. 2, 811--826. https://projecteuclid.org/euclid.rae/1214571369