ContinuityConnected - < f a Proof f a,b is an interval...

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Econ 511 Professor: John Nachbar October 5, 2008 Continuity and Connectedness Theorem 1. Let ( X,d x ) and ( Y,d Y ) be metric spaces. Let f : X Y be continu- ous. Then for any connected set E X , f ( E ) is connected. Proof. I argue by contraposition. Suppose that f ( E ) is separated. I must show that this implies that E is separated. Let F = f ( E ). If F is separated then there are open sets V 1 , V 2 such that, V 1 F 6 = , V 2 F 6 = , F V 1 V 2 and V 1 V 2 = . Let O 1 = f - 1 ( V 1 ) and O 2 = f - 1 ( V 2 ). By continuity, O 1 and O 2 are open. Since, V 1 V 2 = , O 1 O 2 = . Since F V 1 V 2 , E O 1 O 2 . Finally, since V 1 F 6 = , O i E 6 = , for i ∈ { 1 , 2 } . ± The above theorem says that connectedness is a topological property, meaning a property that is preserved by transformation by a continuous function. Compactness is another important example of a topological property. In contrast, convexity is not a topological property. Theorem 2 (Intermediate Value Theorem) . Let [ a,b ] R and let f : [ a,b ] R be continuous. If f ( a ) < f ( b ) then, for any y ( f ( a ) ,f ( b )) , there is an x ( a,b ) such that f ( x ) = y . And an analogous result holds if f ( b )
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Unformatted text preview: < f ( a ) . Proof. f ([ a,b ]) is an interval, since it is connected and since any connected set in R is an interval. Suppose f ( a ) < f ( b ). Since f ( a ) ∈ f ([ a,b ]) and f ( b ) ∈ f ([ a,b ]), it follows that [ f ( a ) ,f ( b )] ⊆ f ([ a,b ]). So, if y ∈ ( f ( a ) ,f ( b )) then y ∈ f ([ a,b ]) and so there is an x ∈ [ a,b ] such that f ( x ) = y . Since y 6 = f ( a ) and y 6 = f ( b ), x ∈ ( a,b ). The argument for f ( b ) < f ( a ) is similar. ± Example 1 . Suppose f ( a ) < 0 and f ( b ) > 0. Then there is an x ∈ ( a,b ) such that f ( x ) = 0. This fact can be used to show the existence of a competitive equilibrium if there are only 2 commodities. To handle the case of more than 2 commodities, more sophisticated machinery must be employed (in particular, the Brouwer fixed point theorem). ² 1...
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This note was uploaded on 05/20/2010 for the course ECON 511 taught by Professor Johnnachbar during the Fall '08 term at Washington University in St. Louis.

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