Unformatted text preview: B is a Borel set, show that f ( B ) is also a Borel set. 4. Let f : R ! R . (a) Suppose f is a monotone function, show that the set of points at which f is discontinuous is countable. (b) Suppose f is any function. Show that the set of points at which f is discontinuous is a Borel set. 5. (a) Prove the Baire Category Theorem: If U 1 , U 2 ,... are open dense subsets of R n , then T 1 i =1 U i is also dense (it may not be open). A subset U of R n is said to be dense if O \ U 6 = ; for every nonempty open set O & R n . (b) Find a Borel set in R that is neither an F & nor a G ± . 1...
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 Fall '10
 GREGORYMARGULIS
 Algebra, Topology, Sets, Empty set, Borel

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