A Continuous Extension that preserves Concavity, Monotonicity and Lipschitz Continuity

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).