Charpter 3 Elements of Point set Topology Open and closed sets in R1 and R2 3.1 Prove that an open interval in R1 is an open set and that a closed interval is a closed set. proof: 1. Let a,b be an open interval in R1, and let x a,b .Consider min x a,b x : L.Then we have B x,L x L,x L a,b .Thatis,x is an interior point of a,b .Sincex is arbitrary, we have every point of a,b is interior.
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Read MoreFunctions of Bounded Variation and Rectifiable Curves Functions of bounded variation 6.1Determine which of the follwoing functions are of bounded variation on 0,1 . (a) f x x2 sin 1/x if x 0, f 0 0. (b) f x x sin 1/x if x 0, f 0 0. Proof: (a) Since f x 2xsin 1/x cos 1/x for x 0,1 and f 0 0,
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