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About differentiability of order one of quasiconvex functions onR n

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This paper is devoted to the study of the different kinds of differentiability of quasiconvex functions onR n. For these functions, we show that Gâteaux-differentiability and Fréchet-differentiability are equivalent; we study the properties of the directional derivatives; and we show that if, for a quasiconvex function, the directional derivatives atx are all finite and two-sided, the function is differentiable atx.

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Communicated by M. Avriel

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Crouzeix, J.P. About differentiability of order one of quasiconvex functions onR n . J Optim Theory Appl 36, 367–385 (1982).

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Key Words

  • Quasiconvex functions
  • Fréchet-differentiability
  • Gâteau-differentiability
  • quasidifferentiability
  • generalized convexity