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

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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 ...
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