1. Since the constraint set is compact, the objective function is continuous in ?, and the solution is unique, the menu is continuous in ?, so (x L (1), t L (1), x H (1), t H (1)) = lim ??1 (x L (?), t L (?), x H (?), t H (?)). Moreover for every 0 < ? < 1, (x L (?), t L (?), x H (?), t H (?)) is nonnegative;U (x L (?), t L (?), p L ) = U (0, 0, p L ), U (x L (?), t L (?), p H ) = U (x H (?), t H (?), p H ), and these properties are preserved in the limit: (x L (1), t L (1), x H (1), t H (1)) is nonnegative, U (x L (1), t L (1), p L ) = U (0, 0, p L ), and U (x L (1), t L (1), p H ) = U (x H (1), t H (1), p H )
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