asn2 - MAT237Y – Multivariable Calculus Summer 2009...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
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 Questions are worth 10 marks each. Due Tuesday, June 2, at 6:10pm sharp. 1. Suppose A ⊂ Rn 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. 2. Define f : Rn → R as f (x) = x 0, 2 sin( 1 x ), if x = 0, if x = 0. Show that f is differentiable everywhere, but that the partial derivatives are not continuous. 3. Suppose f : Rn → R is differentiable at the origin 0 = (0, 0, . . . , 0) (but not necessarily elsewhere), that f (0) = 0, and that c ∈ R is such that ∇f (0) < c. Show that the set U = {x ∈ Rn : f (x) ≤ c x } is a neighbourhood of the origin. Hint: Let ǫ = c − ∇f (0) . No late assignments. 1 ...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online