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Continuity and differentiability properties of monotone real functions of several real variables

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Nonlinear Analysis and Optimization

Part of the book series: Mathematical Programming Studies ((MATHPROGRAMM,volume 30))

Abstract

Continuity and differentiability of monotone functions of several variables are studied; in particular, it is proved that these functions are almost everywhere differentiable. Then it is shown how some properties of Lipschitz functions and quasiconvex functions can be straightforwardly derived from properties of monotone functions.

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References

  1. Y. Chabrillac, “Semi définie positivité de formes quadratiques. Fonctions monotones”, Thèse de 3ème cycle, Université de Clermont (Clermont, France, 1982).

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  2. D.L. Cohn, Measure theory (Birkhauser, Boston, 1980).

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  3. J.-P. Crouzeix, “About differentiability of order 1 of quasiconvex functions,” Journal of Optimization Theory and Applications 35 (1982) 367–385.

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  4. J.-P. Crouzeix, “A review of continuity and differentiability properties of quasiconvex functions on ℝn”, in: J.P. Aubin and R. Vinter, eds., Convex analysis and optimization (Research notes in mathematics 57, Pitman advanced publishing programs, Boston, 1982) pp. 18–34.

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  5. J.-P. Crouzeix, “Continuity and differentiability properties of quasi-convex functions on ℝn, in: S. Schaible and W.T. Ziemba, eds., Generalized concavity in optimization and economics (Academic Press, New York, 1981) pp. 109–130.

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  6. A.J. Ward, “The differential structure of real functions,” Proceedings of the London Mathematical Society 39 (1934) 339–362.

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Author information

Authors and Affiliations

  1. Département de Mathématiques Appliquées, U.E.R. Sciences, Université de Clermont II, B.P. no 45, 63170, Aubière, France

    Yves Chabrillac & J.-P. Crouzeix

Authors
  1. Yves Chabrillac
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  2. J.-P. Crouzeix
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Editor information

B. Cornet V. H. Nguyen J. P. Vial

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© 1987 The Mathematical Programming Society, Inc.

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Chabrillac, Y., Crouzeix, JP. (1987). Continuity and differentiability properties of monotone real functions of several real variables. In: Cornet, B., Nguyen, V.H., Vial, J.P. (eds) Nonlinear Analysis and Optimization. Mathematical Programming Studies, vol 30. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0121151

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  • DOI: https://doi.org/10.1007/BFb0121151

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-00930-3

  • Online ISBN: 978-3-642-00931-0

  • eBook Packages: Springer Book Archive

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