## 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.

### MSC:

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

### Keywords:

Peano derivatives; $$n$$-convex functions