hw2-1 - ECON 831 Homework#2 due in class Oct 15th 1...

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ECON 831 Homework #2 due in class Oct. 15th 1. Consider the self-correspondence Γ on [0 , 1] de ned as follows: Γ( x ) = ( (0 , 1] if x = 0 (0 ,x ) if 0 < x 1 (i) Represent Γ graphically. (ii) Is Γ upper-hemicontinuous? You need to study what happens at each point x [0 , 1] and provide detailed argument/proof. 2. Let X and Y be 2 metric spaces, and Γ 1 : X Y and Γ 2 : X Y 2 correspondences. De ne 2 additional correspondences: Φ : X Y , and Ψ : X Y such that: x X, Φ( x ) = Γ 1 ( x ) Γ 2 ( x ) and Ψ( x ) = Γ 1 ( x ) Γ 2 ( x ) (i) Show that if Γ 1 and Γ 2 are upper-hemicontinuous, then Φ is also upper-hemicontinous. (ii) Assume Γ 1 ( x ) Γ( x ) 6 = for any x X . Show that if Γ 1 and Γ 2 are compact-valued and upper-hemicontinuous, then Ψ is also compact-valued and upper-hemicontinous. 3. Let a be a positive real number, n a natural number, and de ne the subset of R n + as T = [0 ,a ] n = [0 ,a ] × ··· × [0 ,a ] . Let
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This note was uploaded on 02/19/2010 for the course ECON 831 taught by Professor Antoine during the Fall '09 term at Simon Fraser.

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hw2-1 - ECON 831 Homework#2 due in class Oct 15th 1...

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