ans3_graph - Exercise 2: (Upper hemicontinuity) (i)...

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Exercise 2: (Upper hemicontinuity) (i) Consider Γ 1 : R R , Γ 1 ( x ) = (0 ,x 2 + 1] x R . We want to show that Γ 1 is uhc. Consider any real number x R and then Γ 1 ( x ) = (0 ,x 2 + 1] . Consider any open set O R such that (0 ,x 2 + 1] O . Necessarily, there exists δ > 0 s.t. (0 ,x 2 + 1 + δ ) O (NB: since O is an arbitrary open set that contains Γ 1 ( x ) , O could take forms that are a lot more complicated i.e. unions of open sets and such. But no matter how complicated O is, it has to be open and to contain Γ 1 ( x ) .) We need to show that there exists a neighborhood around x s.t. the image of each point in this neighborhood remains in O . Consider ² > 0 [you need to exhibit ² small enough to guarantee the image of the neighborhood does not go outside O ] s.t. ( | x | + ² ) 2 - x 2 < δ 0 < ² < -| x | + x 2 + δ For instance, pick ² = x 2 + δ - p x 2 + δ/ 2 and consider N ², R ( x ) . We have: Γ 1 ( x + ² ) = (0 , ( x + ² ) 2 + 1] and Γ 1 ( x - ² ) = (0 , ( x - ² ) 2 + 1] and we can check that both remain in O . NB: I decided to work with a general x , it might be easier to consider 2 subcases depending on the sign of x . (ii)a) Consider
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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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ans3_graph - Exercise 2: (Upper hemicontinuity) (i)...

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