MATH 3033 Real Analysis Midterm Exam Nov 3 2020 Student Name: Student Id: Tutorial Section: Instructions Please read the following and sign in the blank provided below and sign in the blank below. 1. DO NOT OPEN the exam until you are told to. 2. This is a CLOSED BOOK exam. 3. All mobile phones and communication devices should be switched OFF. 4. You must SHOW YOUR WORK to receive credits in all questions. Answers alone (whether correct or not) will not receive any credit. Question No Points Scores 1 25 2 25 3 25 4 25 Total 100 Integrity Statement I have neither given nor received any unauthorised aid during this exam. The answers submitted are my own work. I understand that sanctions will be imposed if I am found to have violated the University’s regulations governing academic integrity. Signature: 1 1. (25 points) . 1.1 Let P be a point in R2. Prove that R2\{P} is open in R2. 1.2 Let A be compact set in R2, and let B be a closed set in R2. Prove that A ∩B is compact in R2. 1.3 Let An = { ∑n k=−n ak2−k : a−n, a−n+1, · · · , an ∈ {0, 1}}. Show that An is closed in R. Let A = ∪∞n=1An, is A closed? Justify your answer. 2. (25 points) . Let f(x, y) = { x4 x2+y2 , x 2 + y2 6= 0, 0, x2 + y2 = 0 . 1.1 Show that f is continuous on R2; 1.3 Is f differentiable at (0, 0)? Justify your answer and find the derivative if the answer is yes. 3. (25 points) . Consider the following system of two equations x = er + s3 − rs− 1, and y = es − r2 + e2s − 2. Show that near (r, s, x, y) = (0, 0, 0, 0), (r, s) can be expressed as a differentiable function of (x, y), and find the values of ∂r∂x , ∂r ∂y , ∂s ∂x , ∂s ∂x at (x, y) = (0, 0). 4. (25 points) . Consider the equation xy + z ln y + exz = 1. Can you solve the above equation to express one variable in terms of a C1 function of the other two variables near (x, y, z) = (0, 1, 1)? If yes, please list all the possible cases and Justify your answer. Page 2