- Tuesday May 3, 2016
Sets of measure zero. The 1/3 Cantor set. The Cantor function.
Michael Sanchez, Jason Baker, Kyle Henke
Characterization of Riemann Integrable functions: continuous almost everywhere (except on a set of measure zero).
- Thursday May 5, 2016
Completeness of metric spaces.
Justin Campbell, Matt Robinson
Weierstrass Approximation Theorem. (Section 14.8).
Peterson Moyo and Cindi Goodman
- Thursday May 12, 2016
Hanna Butler, Zach Stevens, Marshall Branderburg
Lionel Fiske, Cairn Overturf
Homework 8 (due on Thursday April 28, 2016): Do at least 5 problems
(Exercises 6.1.4 and 6.4.1 count like just one problem).
Exercises 6.1.4 and 6.4.1 (linear transformations are continuous continuous, and continuously differentiable).
Exercise 6.2.2 (uniqueness of derivatives).
Exercise 6.3.1 (differentiable implies directional derivatives).
Exercise 6.3.3 (directional derivatives do not imply differentiable).
Exercise 6.4.3 (chain rule).
Exercise 6.5.1 (non example - Clairaut's Theorem).
Exercise 6.6.7 (Contraction Mapping theorem).
Exercise 6.6.8 (Nearby contractions have nearby fixed points).
Exercise 6.7.3 (application of Inverse Function Theorem).
Reading Assignment: Chapter 6.
Homework 7 (due on Thursday April 14, 2016)
Exercise 4.2.5 and 4.2.6 (polynomials are real analytic. (In 2nd edition there is a typo in the formula in 4.2.5).
Exercise 4.3.1 (summation by parts formula).
Exercise 4.5.1 (basic properties of exponential)
Exercise 4.5.6 (show that the natural logarithm is real analytic for x>0).
Reading Assignment: Chapter 4.
Homework 6 (due on Thursday April 1st, 2016)
Exercise 3.2.2(c) (exploring convergence and uniform convergence of a geometric power series).
Exercises 3.3.4-3.3.5 ( convergence of f_n(x_n) given uniform or pointwise convergence of f_n to f and convergence of x_n to x).
Exercise 3.4.1 and 3.4.3 (the space of continuous functions is complete).
Exercises 3.6.1 and 3.7.3 (interplay between uniform convergence of series, integration and differentiation).
Reading Assignment: Chapter 3.
Homework 5 (due Tuesday March 8, 2016): Do one exercise per section (4 total).
Section 2.1: Exercises 2.1.5 (inclusion map is continuous).
Section 2.1: Exercise 2.1.6 (restriction of a continuous map is continuous).
Section 2.2: Exercise 2.2.2 (prove that the maximum function is continuous only).
Section 2.2: Exercise 2.2.5 (polynomials of two variables are continuous).
Section 2.2: Exercise 2.2.11 (continuity for each variable does not ensure continuity on R^2).
Section 2.3: Exercise 2.3.4 (composition preserves uniform continuity).
Section 2.4: Exercise 2.4.4 (continuity preserves connectedness)
Section 2.4: Exercise 2.4.5 (intermediate value theorem)
Reading assignment: Section 2.2, 2.3, and 2.4.
Homework 4 (due Tuesday March 1st, 2016):
Problems from Book II 3rd edition Chapter 1 (in earlier editions this is
Chapter 12). Do at least 4 of the given exercises.
Exercise 1.1.3 (b)(d) (examples of "sets and pseudometrics"
that fail to be a metric).
Exercise 1.1.5 (Cauchy-Schwarz in R^n and how to use
it to get triangle inequality).
Exercise 1.1.12 (show only that (d) is equivalent to one of the others, because we already showed in class that (a), (b) and (c) are equivalent)
Exercise 1.4.1 (all subsequences of a convergent sequence converge and to the same limit).
Exercise 1.4.3 (convergent sequences are Cauchy sequences).
Exercise 1.5.12 (complete and compact in discrete metric).
Use the open cover property of a compact set to show that the set must be closed and bounded.
Reading Assignment: Book II - Chapter 1.
Homework 3 (Due Tuesday Feb 16, 2016)
Reading Assignment:Section 11.5, 11.6, 11.7 (an example of a bounded function that is not Riemann integrable), and 11.9 (Fundamental Theorems of calculus).
- Section 11.4: Exercise 11.4.2 (if f is continuous, positive, and Riemann
integrable, and it integral is zero then f must be zero).
- Section 11.6: Exercise 11.6.3 (integral test for series).
- Section 11.9: Exercise 11.9.2 (two antiderivatives of the same function must be equal up to a constant).
Homework 2 (Due on Thursday Feb 4th, 2016)
Reading Assignment Sections 11.3, 11.4
- Section 11.3: Exercise 11.3.5 (show only that upper Riemann integral is equal to the infimum of the upper Riemann sums).
- Section 11.4: Exercise 11.4.1 (a) and (d) only (integration laws: linearity and precursor of monotonicity).
Homework 1 (due on Tue 1/26/2016):
Exercises from Tao's Book I (hardcover edition), Chapter 11 on Riemann Integration:
Reading Assignment Sections 11.1, 11.2 and 11.3
- Exercise 11.1.2 (intersection of two bounded intervals is
a bounded interval),
- Exercise 11.1.4 (P1#P2 is a partition and a common refinement for both P1 and P2)
- Exercise 11.2.2 (only show if f,g are piecewise constant
(p.c.) on I then max(f,g) is p.c. on I),
- Exercise 11.2.4(g)(h) (laws of p.c. integration).
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Last updated: January 29, 2016