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# Q1mat237 -...

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1. (a) [4 marks] Prove that the set S = in R } 1 : ) , {( 2 2 y x y x + < 2 is open. You have to give a rigorous proof based on the definition of an open set. (b) [2 marks] Give an example of a set S such that the interior of S is unequal to the interior of the closure of S. . Justify your answer. 2

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2. (a) [3 marks] Show directly from the definition that the set } 1 : ) , {( 2 = = xy R y x S is disconnected. That is produce a disconnection (in a set notation) for S. (b) [3 marks] Suppose S 1 and S 2 are connected sets in R n that contain at least one point in common. Is it true that S 1 S 2 must be connected? Justify. 3
3. (a) [5 marks] Determine whether 4 4 4 5 ) 0 , 0 ( ) , ( 2 lim y x xy x y x + exists, and if so, find its value. (b) [1 mark] Fill the blank: If f : R m R n is defined on the set U R m , then f is continuous at the point a U if ε > 0 . ..……………. .....................

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Unformatted text preview: ………………………………………………………………………….” 4 4. (a) [1 mark] Formulate “The Monotone Sequence Theorem” (b) [5 marks] Show that exists and find its value if is defined k k x +∞ → lim } { k x recursively by 1 = x and 2 1 2 1 1 + = + k k x x 5 5. (a) [2 marks] Let denote the line segment in R i L 2 from the origin (0, 0) to the point ) 1 , 1 ( 2 i i , ( i = 1, 2, 3, …. .) on the curve . Is the union 2 ) ( x x f = S = compact? Justify your answer. i i L ∞ = 1 U (b) [4 marks] Suppose S ⊂ R n is compact, f : S R is continuous, and f ( x ) > 1 for → every x ∈ S . Show that there is a number c > 1 such that f ( x ) ≥ c for every x ∈ S . 6...
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Q1mat237 -...

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