MATH2330: ANALYSIS|ASSIGNMENT Name: Student No: Due: In

MATH2330: ANALYSIS|ASSIGNMENT Name: Student No: Due: In class at 9am Friday 10th May 2013 (Week 9). Marking: As one of the objectives of this course is to teach students how to communicate mathe- matics clearly I expect you to provide clear and careful explanations in your answers. Hence marks will be allocated for exposition as well as mathematical content; twenty percent of your mark will be allocated for exposition and eighty percent for mathematical content. All questions are worth an equal number of marks. You are required to complete this sheet (name and student number) and attach it to your worked so- lutions. If you use LATEX please place the solution of each question directly after the stated problem. Problem 1. Suppose that a function f(x) dened on [0; 1] satises f(1=n) ! 0 as n ! 1. Is it true that f(x) ! 0 as x ! 0+ provided (a) f is continuous on [0; 1] ? (b) f is dierentiable (0; 1) ? Problem 2. Show that the equation 2x = 2 ?? x2 has at least two real roots. Problem 3. Show that the function f(x) = sin x is continuous on R. Is it uniformly continuous on R? Problem 4. Is the function f(x) =8