# a1f08 - s compact Does it follow that the set S is compact...

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ASSIGNMENT #1 1. Give an example (in the set notation, not a picture) of a closed set S in 2 R such that S S int (that is the closure of is not equal to S ). int S 2. Let S be a set in n R . Is it true that every interior point of S is in ? Justify. int S 3. Prove that the set in } 1 : ) , {( > = y y x S 2 R is open. You have to give a rigorous proof based on the definition of an open set. 4. Consider the sequence in )} sin ), 1 {(arctan( } { 2 k k k + = x 2 R . Is there a convergent subsequence? Justify your answer. 5. Let denote the line segment in i L 2 R between the points ) / 1 , / 1 ( i i and with . Is the union compact? Justify your answer. ) / 1 , / 1 ( i i ... , 3 , 2 , 1 = i i i L S = = 1 U 6. Prove or disprove the statement “If a set S n R is connected, then its interior is also connected” int S 7. Consider the function defined by Is there an absolute maximum value of f on the set R R f 2 : . ) , ( x y e y x f y x + = + } 2 : ) , {( + = y x y x S ? Justify. 8. Suppose that the image of the set S under the continuous map
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Unformatted text preview: s compact. Does it follow that the set S is compact? Justify your answer. R R S f n → ⊂ : i 9. Is there a constant such that the function R c ∈ ⎪ ⎩ ⎪ ⎨ ⎧ = ≠ + + = ) , ( ) , ( ) , ( ) , ( ) , ( 2 2 3 y x for c y x for y x y xy y x f is continuous at (0, 0)? Justify your answer. 10. Evaluate the following limits or prove that they do not exist (a) ) 1 , 1 ( ) , ( lim → y x 4 3 4 y x y x − − (b) ) , ( ) , ( lim → y x 2 2 2 2 2 ) ( ) cos( 1 y x y x + + − (c) ) , , ( ) , , ( lim → z y x 2 2 2 z y x xz yz xy + + + + Knowing: all the definitions, precise formulation of all the Theorems done on the lecture and the proofs of the few Theorems listed below is also part of the Assignment. 1.4, 1.13, 1.14, 1.22, 1.26....
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