## Real Analysis Exchange

### Infinite Peano Derivatives

Miklós Laczkovich

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)$cannot be residual and of full measure simultaneously. #### Article information Source Real Anal. Exchange, Volume 26, Number 2 (2000), 811-826. Dates First available in Project Euclid: 27 June 2008 Permanent link to this document https://projecteuclid.org/euclid.rae/1214571369 Mathematical Reviews number (MathSciNet) MR1844395 Zentralblatt MATH identifier 1012.26004 #### Citation Laczkovich, Miklós. Infinite Peano Derivatives. Real Anal. Exchange 26 (2000), no. 2, 811--826. https://projecteuclid.org/euclid.rae/1214571369 #### References • S. Agronsky, A. M. Bruckner, M. Laczkovich and D. Preiss, Convexity conditions and intersections with smooth functions, Trans. Amer. Math. Soc. 289 (1985), 659-677. • P. S. Bullen, A criterion of$n$-convexity, Pacific J. Math. 36 (1971), 81–98. • P. S. Bullen and S. N. Mukhopadhyay, Relations between some general$n\$th-order derivatives, Fund. Math. 85 (1974), 257–276.
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