MATH 105 Berkeley

Mathematics 105, Spring 2004 - M. Christ Final Exam Review Guide The final exam will primarily emphasize the portion of the course concerned with Lebesgue integration, in which we followed Stroock\'s Chapters 2, 3, 4.1 and 5.0,2,3. From the first part

Mathematics 105 - Spring 2004 - M. Christ A Supplementary Note, Selected Solutions for Problem Set 10, and Problem Set 11 Proposition. Denote Lebesgue measure in Rk by k , and as usual let BRk , B Rk denote the classes of Borel and Lebesgue measurabl

Mathematics 105 Spring 2004 Problem Set 1 The first topic of this course is differentiation of functions of several variables, culminating in the inverse and implicit function theorems. Our test will be Chapters 1 and 2 of Spivak\'s Calculus on Manifo

Mathematics 105 - Spring 2004 - M. Christ Problem Set 2 - Due Friday February 6 Solve the following problems from Spivak Chapter 2: 1,5,7,8,10(f),13,15,16,18(c),20(a),22,24. Comments: #5: You\'re asked to show that the given function is not differenti

Mathematics 105, Spring 2004 - M. Christ Final Exam Solutions (selecta) 1 (2e) True or false: If K R is a compact set of Lebesgue measure zero, and if f : R R is a homeomorphism (that is, f is continous and invertible, and f -1 is also continuous),

Mathematics 105, Spring 2004 - Midterm Exam #1 Comments (1b) Give an example of a function f : R3 R1 such that f is not differentiable at a = (0, 0, 0), but all partial derivatives Di f (a) do exist. Comment: A common answer was f (x, y, z) = xyz/ x

Mathematics 105, Spring 2004 - Midterm Exam #1 Solutions (1b) Give an example of a function f : R3 R1 such that f is not differentiable at a = (0, 0, 0), but all partial derivatives Di f (a) do exist. Solution: Define f (0, 0, 0) = 0, and for all x

Mathematics 105, Spring 2004 - Problem Set IV Solutions 1 IV.A Let {In } be any finite set of open intervals that covers [0, 1] Q. Show that n |In | 1. Explain why this does not prove that |[0, 1] Q|e 1. Solution. This does not prove that |[0,

Mathematics 105 - Spring 2004 - M. Christ Problem Set 9 - Solutions to Selecta IX.A Consider the measure space (R1 , B R1 , ) where denotes Lebesgue measure. 1 Consider the measurable functions fn (x) = n [0,n] . Show that fn 0 uniformly on R. Show