A Continuous Extension that Preserves Concavity, Monotonicity and Lipschitz Continuity
The following is proven here: let W : X Ã C −→ R, where X is convex, be a continuous and bounded function such that for each y ∈ C, the function W (Â·, 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 Ã Y −→ Â£infXÃC W, supXÃC W Â¤, such that for each y ∈ Y , the function U (Â·, y) : X −→ R is concave (resp. strongly concave; resp. Lipschitzian with constant My ; resp. monotone; resp. strictly monotone).
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- Howe, Roger, 1987. "Sections and extensions of concave functions," Journal of Mathematical Economics, Elsevier, vol. 16(1), pages 53-64, February.
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