Math 521Exam I One HourFriday March 5, 2010I. (25 points) Assume X ⊂ Rm, Y ⊂ Rn, f : X → Y , U ⊂ Rm. Completethe following definitions:• The map f is continuous iff• The map f is uniformly continuous iff• The map f is lipschitz on X iff• The set U is open iff1II (25 points) A theorem we proved in class says that subset S of R is closed ifand only if it is closed under limits of sequences. State this theorem preciselyand prove it. Include the relevant definitions.2III (25 points) Prove that the functionH(x) =1 if x ≥ 00 if x < 0is not continuous. Use the continuity definition you gave on question I.3IV (25 points) Let f : X → Y , A ⊂ X, B ⊂ Y . True or false? (Giveproof if true, counter example if false.)• f(f−1(B)) ⊆ B• B ⊆ f (f−1(B))• f−1(f(A)) ⊆ A• A ⊆
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