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# ps7 - x → ∞ f x and lim x →-∞ f x both exist and...

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18.100B and 18.100C Fall 2011 Problem Set 7 Due November 3rd at 4 pm in room 2-108. Hand in parts 1, 2 and 3 separately. Put your name and whether you are registered for 18.100B or 18.100C on each part. Part 1 1. Let X and Y be metric spaces and f : X Y a function. Prove that f is continuous if and only if f ( E ) f ( E ) for any subset E X . 2. Problem 10 from page 99. Part 2 3. Problem 14 from page 100. 4. Let f : R R be continuous and suppose that
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Unformatted text preview: x → + ∞ f ( x ) and lim x →-∞ f ( x ) both exist and are ﬁnite. Prove that f is uniformly continuous. Part 3 5. Let K be a compact metric space with metric d and suppose f : K → K is distance preserving , meaning that d ( f ( x ) ,f ( y )) = d ( x,y ) for all x,y ∈ K . Prove that f ( K ) = K . 6. Problem 1 from page 114. 1...
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