{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# asn2-soln - MAT237Y – Multivariable Calculus Summer 2009...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}

### Page1 / 3

asn2-soln - MAT237Y – Multivariable Calculus Summer 2009...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online