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00fall-midterm

00fall-midterm - F = f ◦ g is continuous at x 5 Let D =...

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MAT 371 Advanced Calculus / 100 Oct 26, 2000 Mid-term exam name 1 . State (complete) the “standard” definitions for: a. An infinite sequence ( a n ) n ZZ + is a Cauchy sequence if . . . b. Suppose S IR and z IR. Then z is an accumulation point of S if . . . c. Suppose D IR and f : D IR. The function f is uniformly continuous if . . . 2. a . State the Heine-Borel theorem. b . State two theorems which guarantee that a sequence a = ( a n ) n ZZ + converges, 3. For each of the following sets S i IR find the set S i of all accumulation points of S i and the closure S i . (No proofs required). a . S 1 = (0 , ) b . S 2 = Q (rational numbers) c . S 3 = ZZ (integers) d . S 4 = { 2 n : n ZZ } . ( Caution: n may be negative or positive .) 4. Suppose D, E IR, g : D E is continuous at x 0 D and f : E IR is continuous at y 0 = g ( x 0 ). Prove that
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Unformatted text preview: F = f ◦ g is continuous at x . 5. Let D = IR \ { } and f : D 7→ IR be defined by f ( x ) = 1 x . a. Is f continuous? Prove that your answer is correct! b. Is f uniformly continuous? Prove that your answer is correct! ( In either part you may use any theorem proved in class or in the homework ). 6. Let Q be the set of rational numbers and let f : Q 7→ IR be defined by f ( x ) = 1 q if x = p q with p,q ∈ ZZ, q > 0 and gcd( p,q ) = 1. For each of x 1 = 2 3 and x 2 = √ 2 either prove that lim x-→ x i f ( x ) does not exist, or calculate the limit and prove that it is indeed the limit....
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