# zbMATH — the first resource for mathematics

The Denjoy-Clarkson property with respect to Hausdorff measures for the gradient mapping of functions of several variables. (English) Zbl 1154.26016
Summary: We construct a differentiable function $$f:\mathbb{R}^{^{ n }}\rightarrow \mathbb{R}$$ $$(n\geq 2)$$ such that the set $$(\nabla f)^{-1}(B(0,1))$$ is a nonempty set of Hausdorff dimension 1. This answers a question posed by Z. Buczolich [Rev. Mat. Iberoam. 21, 889–910 (2005; Zbl 1116.26007)].

##### MSC:
 26B05 Continuity and differentiation questions 28A75 Length, area, volume, other geometric measure theory
Full Text:
##### References:
 [1] Bruckner, A., Differentiation of real functions, 5, (1994), American Mathematical Society, Providence, RI · Zbl 0796.26004 [2] Buczolich, Z., The $$n$$-dimensional gradient has the 1-dimensional Denjoy-clarkson property, Real Anal. Exchange, 18, 221-224, (199293) · Zbl 0783.26010 [3] Buczolich, Z., Another note on the gradient problem of C. E. Weil, Real Anal. Exchange, 22, 775-784, (199697) · Zbl 0940.26011 [4] Buczolich, Z., Solution to the gradient problem of C. E. Weil, Rev. Mat. Iberoamericana, 21, 889-910, (2005) · Zbl 1116.26007 [5] Clarkson, J. A., A property of derivatives, Bull. Amer. Math. Soc., 53, 124-125, (1947) · Zbl 0032.27102 [6] Denjoy, A., Sur une proprieté des fonctions dérivées, Enseignement Math., 18, 320-328, (1916) · JFM 46.0381.05 [7] Deville, R.; Matheron, É., Infinite games, Banach space geometry and the eikonal equation, Proc. Lond. Math. Soc. (3), 95, 1, 49-68, (2007) · Zbl 1163.91007 [8] Engelking, R., General Topology, (1989), Heldermann Verlag, Berlin · Zbl 0684.54001 [9] Holický, P.; Malý, J.; Weil, C. E.; Zajíček, L., A note on the gradient problem, Real Anal. Exchange, 22, 225-235, (199697) · Zbl 0879.26041 [10] Malý, J., The Darboux property for gradients, Real Anal. Exchange, 22, 1, 167-173, (199697) · Zbl 0879.26042 [11] Malý, J.; Zelený, M., A note on buczolich’s solution of the Weil gradient problem: a construction based on an infinite game, Acta Math. Hungar., 113, 145-158, (2006) · Zbl 1127.26006 [12] Mattila, P., Cambridge Studies in Advanced Mathematics, 44, Geometry of sets and measures in Euclidean spaces, (1995), Cambridge University Press, Cambridge · Zbl 0819.28004 [13] Weil, C. E., Query 1, Real Anal. Exchange, 16, 373 pp., (199091)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.