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A Continuous Extension that preserves Concavity, Monotonicity and Lipschitz Continuity

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Abstract

The following is proven here: let W : X × C ? R, where X is convex, be a continuous and bounded function such that for each y?C, the function W (·,y) : X ? R is concave (resp. strongly concave; resp. Lipschitzian with constant M; resp. monotone; resp. strictly monotone) and let Y?C. If C is compact, then there exists a continuous extension of W, U : X × Y ? [infX×C W,supX×C W], such that for each y?Y, the function U(·,y) : X ? R is concave (resp. strongly concave; resp. Lipschitzian with constant My; resp. monotone; resp. strictly monotone).

Suggested Citation

  • Andrés Carvajal, 2004. "A Continuous Extension that preserves Concavity, Monotonicity and Lipschitz Continuity," Royal Holloway, University of London: Discussion Papers in Economics 04/27, Department of Economics, Royal Holloway University of London, revised Nov 2004.
    • Handle: RePEc:hol:holodi:0427
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    2. Andr�s Carvajal, 2003. "Testable Restrictions og General Equilibrium Theory in Exchange Economies with Externalities," Borradores de Economia 3556, Banco de la Republica.

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