Unformatted text preview: ρ E the distance function from a point x ∈ X to E : ρ E ( x ) = inf y ∈ E d ( x, y ) . Prove that ρ E is uniformly continuous. 4. Let ( X, d X ) and ( Y, d Y ) be metric spaces, and f : X → Y a continuous function. If X is compact, prove that f is uniformly continuous. 5. Is it true that if f : R k → R k is continuous and A ⊂ R k is convex, then f ( A ) is convex? 6. Consider a normed space ( X, k · k ). Prove that the closure of an open ball is the closed ball, i.e., { x ∈ X k x k < R } = { x ∈ X k x k ≤ R } . 1...
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This note was uploaded on 12/27/2011 for the course MATH 118a taught by Professor Stopple,j during the Fall '08 term at UCSB.
 Fall '08
 Stopple,J
 Integers

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