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# Zeitschrift für Analysis und ihre Anwendungen

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**Volume 21, Issue 3, 2002, pp. 691–707**

**DOI: 10.4171/ZAA/1103**

Published online: 2002-09-30

Level Sets of Hölder Functions and Hausdorff Measures

Emma D'Aniello^{[1]}(1) Università degli Studi di Napoli, Caserta, Italy

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.

*Keywords: *Level sets, $C^{m, \alpha}$ functions

D'Aniello Emma: Level Sets of Hölder Functions and Hausdorff Measures. *Z. Anal. Anwend.* 21 (2002), 691-707. doi: 10.4171/ZAA/1103