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Unformatted text preview: X . Let Z ( f ) (the zero set of f ) be the set of all p ∈ X at which f ( p ) = 0. Prove that Z ( f ) is closed. 8. Let f and g be continuous mappings of a metric space X into a metric space Y , and let E be a dense subset of X . Prove that f ( E ) is dense in f ( X ). If g ( p ) = f ( p ) for all p ∈ E , prove that g ( p ) = f ( p ) for all p ∈ X . 9. Let I = [ a, b ] a closed interval. Suppose f : I → I is continuous. Prove that there must be at least one point x ∈ [ a, b ] such that f ( x ) = x . 10. Let X be a metric space, Y a complete metric space, and E ⊂ X a dense set. Suppose that f : E → Y is uniformly continuous. Prove that there exists a unique continuous extension of f to the whole space X , i.e., ∃ ! g : X → Y continuous such that f ( x ) = g ( x ) for all x ∈ E ....
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- Fall '08