Draw the following sets and determine if they are open or closed, bounded or unbounded: A...

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Draw the following sets and determine if they are open or closed, bounded or unbounded: A₁ = {(x, y) = R²:|r1| + y + 3) ≤ 2} = {(x, y) = R²: log(a + 1) - y ≥ 0 ^ly +3|<1} A3 = {(x, y) = R²: x² + 16y² <16}. Hint: To prove that a set is open or closed, use that, given a continuous function, the preimage of an open set is open and the preimage of a closed set is closed. Exercise 2. Determine the interior and the closure of the following two sets (a) S = {(x,y) = R²|0<√²+² <1} (2.5 P.) (b) S = {(A) In € N} (2.5 P.) Exercise 3. Determine whether the two functions fi fa: RxR-R (a) (2.5 P.) (b) (2.5 P.) fi(x,y)= J2(x,y) = { 1 " . for (x,y) (0,0) for (r. y) = (0,0) for (x,y) = (0,0) for (x,y) = (0.0) are continuous at (x, y) = (0,0). Hint: For fi find a suitable sequence and study its convergence. Draw the following sets and determine if they are open or closed, bounded or unbounded: A₁ = {(x, y) = R²:|r1| + y + 3) ≤ 2} = {(x, y) = R²: log(a + 1) - y ≥ 0 ^ly +3|<1} A3 = {(x, y) = R²: x² + 16y² <16}. Hint: To prove that a set is open or closed, use that, given a continuous function, the preimage of an open set is open and the preimage of a closed set is closed. Exercise 2. Determine the interior and the closure of the following two sets (a) S = {(x,y) = R²|0<√²+² <1} (2.5 P.) (b) S = {(A) In € N} (2.5 P.) Exercise 3. Determine whether the two functions fi fa: RxR-R (a) (2.5 P.) (b) (2.5 P.) fi(x,y)= J2(x,y) = { 1 " . for (x,y) (0,0) for (r. y) = (0,0) for (x,y) = (0,0) for (x,y) = (0.0) are continuous at (x, y) = (0,0). Hint: For fi find a suitable sequence and study its convergence.

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Exercise 1 a A xy R y r 2 is a closed bounded set To see why note that the function fxy r1y3 is cont... View the full answer