Revista Matemática Iberoamericana

Solution to the gradient problem of C.E. Weil

Zoltán Buczolich

Full-text: Open access


In this paper we give a complete answer to the famous gradient problem of C. E. Weil. On an open set $G\subset \mathbb{R}^{2}$ we construct a differentiable function $f:G\to\mathbb{R}$ for which there exists an open set $\Omega_{1}\subset\mathbb{R}^{2}$ such that $\nabla f(\mathbf{p})\in \Omega_{1}$ for a $\mathbf{p}\in G$ but $\nabla f(\mathbf{q})\not\in\Omega_{1}$ for almost every $\mathbf{q}\in G$. This shows that the Denjoy-Clarkson property does not hold in higher dimensions.

Article information

Rev. Mat. Iberoamericana, Volume 21, Number 3 (2005), 889-910.

First available in Project Euclid: 11 January 2006

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 26B05: Continuity and differentiation questions
Secondary: 28A75: Length, area, volume, other geometric measure theory [See also 26B15, 49Q15] 37E99: None of the above, but in this section

gradient Denjoy-Clarkson property Lebesgue measure


Buczolich, Zoltán. Solution to the gradient problem of C.E. Weil. Rev. Mat. Iberoamericana 21 (2005), no. 3, 889--910.

Export citation


  • Brucks, K.M. and Buczolich, Z.: Trajectory of the turning point is dense for a co-$\sigma$-porous set of tent maps. Fund. Math. 165 (2000), no. 2, 95-123.
  • Buczolich, Z.: The $n$-dimensional gradient has the $1$-dimensional Denjoy-Clarkson property. Real Anal. Exchange, 18 (1992/93), no. 1, 221-224.
  • Buczolich, Z.: Level sets of functions $f(x,y)$ with nonvanishing gradient. J. Math. Anal. Appl., 185 (1994), no. 1, 27-35.
  • Buczolich, Z.: Approximate continuity points of derivatives of functions of several variables. Acta Math. Hungar. 67 (1995), no. 3, 229-235.
  • Buczolich, Z.: Another note on the gradient problem of C.E. Weil. Real Anal. Exchange, 22 (1996/97), no. 2, 775-784.
  • Buczolich, Z.: Functions of two variables with large tangent plane sets. J. Math. Anal. Appl., 220 (1998), no. 2, 562-570.
  • Clarkson, J.A.: A property of derivatives. Bull. Amer. Math. Soc. 53 (1947), 124-125.
  • Denjoy, A.: Sur une proprieté des fonctions dérivées. Enseignement Math. 18 (1916), 320-328.
  • Federer, H.: Geometric measure theory. Die Grundlehren der matematischen Wissenschaften 153. Springer-Verlag, New York, 1969.
  • Holický, P., Malý, J., Zaj\' \iček, L. and Weil, C.E.: A note on the gradient problem. Real Anal. Exchange, 22 (1996/97), no. 1, 225-235.
  • Petruska, G.: Derivatives take every value on the set of approximate continuity points. Acta Math. Hungar. 42 (1983) no. 3-4, 355-360.
  • Weil, C.E.: On properties of derivatives. Trans. Amer. Math. Soc. 114 (1965), 363-376.
  • Query 1. Real Anal. Exchange 16 (1990/91), no. 1, 373.