204Lecture72009web - Economics 204 Lecture 7Tuesday, August...

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Economics 204 Lecture 7–Tuesday, August 4, 2009 Note : In this set of lecture notes, ¯ A refers to the closure of A . Section 2.9, Connected Sets Defnition 1 Two sets A, B in a metric space are separated if ¯ A B = A ¯ B = A set in a metric space is connected if it cannot be written as the union of two nonempty separated sets. Example: [0 , 1) and [1 , 2] are disjoint but not separated: [0 , 1) [1 , 2] = [0 , 1] [1 , 2] = { 1 } 6 = [0 , 1) and (1 , 2] are separated: [0 , 1) (1 , 2] = [0 , 1] (1 , 2] = [0 , 1) (1 , 2] = [0 , 1) [1 , 2] = Note that d ([0 , 1) , (1 , 2]) = 0 even though the sets are separated. Note that separation does not require that ¯ A ¯ B = . [0 , 1) (1 , 2] is not connected. Theorem 2 (9.2) Ase t S of real numbers is connected if and only if it is an interval, i.e. given x, y S and z ( x, y ) ,then z S . ProoF: First, we show that S connected implies that S is an interval. We do this by proving the contra- positive: if S is not an interval, it is not connected. If S is not an interval, ±nd x, y S, x < z < y, z 6∈ S 1
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Let A = S ( −∞ ,z ) ,B = S ( z, ) Then ¯ A B ( −∞ ) ( )=( −∞ ] ( )= A ¯ B ( −∞ ) ( −∞ ) [ A B =( S ( −∞ )) ( S ( )) = S \{ z } = S x A, so A 6 = y B, so B 6 = so S is not connected. We have shown that if S is not an interval, then S is not connected; therefore, if S is connected, then S is an interval. Now, we need to show that if S is an interval, it is connected. This is much like the proof of the Intermediate Value Theorem. See de la Fuente for the details. Theorem 3 (9.3) Let X be a metric space, f : X Y continuous. If C is a connected subset of X ,then f ( C ) is connected. Proof: This is problem 5(b) on Problem Set 3. The idea is in the diagram. Prove the contrapositive: if f ( C ) is not connected, then C is not connected. Corollary 4 (Intermediate Value Theorem) If f :[ a, b ] R is continuous, and f ( a ) <d<f ( b ) , then there exists c ( a, b ) such that f ( c d . 2
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Proof: This is our third, and slickest, proof of the Intermediate Value Theorem. It is short because a substantial part of the proof was incorporated into the proof that C R is connected if and only if C is an interval, and the proof that if C is connected, then f ( C ) is connected. Here’s the proof: [ a, b ]is an interval, so [ a, b ] is connected, so f ([ a, b ]) is connected, so f ([ a, b ]) is an interval. f ( a ) f ([ a, b ]), and f ( b ) f ([ a, b ]), and d [ f ( a ) ,f ( b )]; since f ([ a, b ]) is an interval, d f ([ a, b ]), i.e. there exists c [ a, b ] such that f ( c )= d .S ince f ( a ) <d<f ( b ), c 6 = a , c 6 = b ,so c ( a, b ).
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204Lecture72009web - Economics 204 Lecture 7Tuesday, August...

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