# Level Sets of Hölder Functions and Hausdorff Measures

### Emma D'Aniello

Università degli Studi di Napoli, Caserta, Italy

## Abstract

In this paper we investigate some connections between Hausdorff measures, Hölder functions and analytic sets in terms of images of zero-derivative sets and level sets. We characterize in terms of Hausdorff measures and descriptive complexity subsets $M \subseteq \mathbb R$ which are

(1) the image under some $C^{m, \alpha}$ function $f$ of the set of points where the derivatives of first $n$ orders are zero

(2) the set of points where the level sets of some $C^{m, \alpha}$ function are perfect

(3) the set of points where the level sets of some $C^{m, \alpha}$ function are uncountable.

## Cite this article

Emma D'Aniello, Level Sets of Hölder Functions and Hausdorff Measures. Z. Anal. Anwend. 21 (2002), no. 3, pp. 691–707

DOI 10.4171/ZAA/1103