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Unformatted text preview: ∈ R m and a diﬀerentiable function g : R ‘ → R m such that, for any x λ ∈ R ‘ , f ( x λ ,g ( x λ )) = y . For any x λ ∈ R ‘ , if D μ f ( x ) has full rank then Dg ( x λ ) =[ D μ f ( x )]1 D λ f ( x ) . 5. (a) Let X = { , 1 } . ( X consists of just two points.) Make X a metric space by giving it the standard Euclidean metric. Prove that the set { } is both open and closed in this metric space. (b) Let X = R n . The boundary of A ⊂ R n , written ∂A , is deﬁned to be ∂A = A ∩ A c . Prove that if both A and A c are nonempty then ∂A 6 = ∅ . ( Hint. Fix two points a ∈ A and b ∈ A c . Show that there is a t such that x t = ta + (1t ) b ∈ ∂A .) (c) Again, let X = R n . Prove that if A and A c are both nonempty then A cannot be both open and closed....
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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.
 Fall '08
 JohnNachbar
 Economics

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