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Unformatted text preview: R m and a dierentiable function g : R R m such that, for any x R , f ( x ,g ( x )) = y . For any x R , if D f ( x ) has full rank then Dg ( x ) =[ D f ( x )]1 D 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 dened 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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 Fall '08
 JohnNachbar
 Economics

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