Real Analysis Exchange

The Darboux property for gradients

Jan Malý

Full-text: Open access

Abstract

It is well known that the derivative of a function of one variable has the Darboux property. In this paper it is shown that the gradient of a differentiable function of several variables maps certain closed convex sets to connected sets.

Article information

Source
Real Anal. Exchange, Volume 22, Number 1 (1996), 167-173.

Dates
First available in Project Euclid: 1 June 2012

Permanent link to this document
https://projecteuclid.org/euclid.rae/1338515211

Mathematical Reviews number (MathSciNet)
MR1433604

Zentralblatt MATH identifier
0879.26042

Subjects
Primary: 26B05: Continuity and differentiation questions

Keywords
Darboux property differentiable gradient

Citation

Malý, Jan. The Darboux property for gradients. Real Anal. Exchange 22 (1996), no. 1, 167--173. https://projecteuclid.org/euclid.rae/1338515211


Export citation

References

  • A.,M. Bruckner, Differentiation of Integrals, Supplement to Amer. Math. Monthly 78,9, Part II (1971).
  • J. Dieudonné, Foundations of Modern Analysis, Academic Press, New York and London 1969.
  • L. Mišik, Der Mittelwertsatz für additive Intervallfunktionen, Fund. Math. 45 (1957), 64-70.
  • C.,J. Neugebauer, Darboux property for functions of several variables, Trans. Amer. Math. Soc. 107 (1963), 30-37.
  • R. R. Phelps, Convex Functions, Monotone Operators and Differentiability, Lecture Notes in Math. 1364, Springer-Verlag, Berlin 1993.
  • C.,E. Weil, A topological lemma and applications to real functions, Pacific J. Math. 44,2 (1973), 757-765.