18.100B.Homework7

# 18.100B.Homework7 - 18.100B/C Fall 2008 Homework 7...

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Unformatted text preview: 18.100B/C: Fall 2008 Homework 7 Available Monday, October 20 Due Wednesday, October 29 1. Let f : X → Y be a continuous function between metric spaces. Define a function δ f : X × (0 , ∞ ) → (0 , ∞ ) as follows: δ f ( x, ǫ ) = sup { δ > 0 ; ∀ t ∈ X d X ( x, t ) < δ ⇒ d Y ( f ( x ) , f ( t )) < ǫ } . (a) (5pts) Show that the statement “ f is continuous at x ” is equivalent to ” δ f ( x, ǫ ) > for each ǫ > ”. (b) (5pts) Show that f is uniformly continuous on X iff inf x ∈ X δ f ( x, ǫ ) > . [ Hint : you need to find a uniform δ > ; choose it to be this infimum.] (c) (5pts) Consider the function f ( x ) = x 2 defined on the metric space X = [0 , ∞ ) . Show that, for each x ∈ X , sup t ∈ X,d X ( x,t ) <δ | f ( x ) − f ( t ) | = 2 xδ + δ 2 . (d) (5pts) Use part (c) to show that δ f ( x, ǫ ) = √ x 2 + ǫ − x . Show that, for fixed ǫ > , lim x →∞ [ √ x 2 + ǫ − x ] = 0 . Conclude that f is not uniformly continuous on X . [ Hint : you can use Calculus here, but you needn’t. Setyou can use Calculus here, but you needn’t....
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18.100B.Homework7 - 18.100B/C Fall 2008 Homework 7...

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