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Unformatted text preview: MAT237Y Multivariable Calculus Summer 2009 Assignment # 2 solutions. Questions are worth 10 marks each. 1. Suppose A R n is not compact. (a) Show there is a continuous function f : A R such that f ( A ) is not bounded. (b) Show there is a continuous function g : A R such that g ( A ) is not closed. Proof. (a) Since A R n is not compact, then either A is unbounded or not closed. In the first case we can choose f ( x ) = bardbl x bardbl then f is continuous and f ( A ) is unbounded. In the latter case there exists y R n \ A and y i A ( i = 1 , 2 , ) such that y i y as i . Now let f ( x ) = 1 bardbl x y bardbl , then f is a well-defined continuous function on A and f ( A ) is unbounded. (b) If A is unbounded we choose g ( x ) = 1 1 + bardbl x bardbl . Now g is continuous. But 0 / g ( A ) is a limit point of g ( A ), so g ( A ) is not closed. If A is not closed we can find y and y i s as above. Let g ( x ) = bardbl x y bardbl , then...
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