Real Analysis Homework 2002

Measure and Integration
Fall 2015 - Fun Page

Week 1 -- "Due" Thursday, September 17

Work Sheet 1

For one of the following write a paragraph or two explaining your method of solution and in what sense it solves the problem. Of course, give your solution too!!

A dart is thrown randomly at a dart board ( English Competition Board) What is the probability of hitting the red . Note, you can print the picture of the board and measure the ring sizes from the picture. Since you want probability, actual measurements are not important, only relative measurements -- so you can use the picture!
The problem is that nobody wants a nuclear dump in their backyard. But Minnesota needs one, so to be fair, the state legislature (in their collective wisdom) has decided to randomly choose a location somewhere in Minnesota and build the dump there. However, they also instruct the mathematicians at St. Olaf to "determine the probability that the site chosen will be within 100 miles of a major metropolitan area." Estimate this probability.

Problem Set 1

State and prove both versions of the FTC.
Prove that every differentiable function is continuous.
Find a derivative which is not continuous. (Inherent here is that you not only give the function and find its derivative, but prove the derivative is discontinuous at some point.)
Find a derivative which is not continuous at 100 points.
Find a derivative which is not continuous at infinitely many points.
Let C₁₀={.d₁d₂d₃... : d_i= 0 or 9 for each i}. Determine the "geometry" of C₁₀. ie locate portions of R which contain points from C and portions devoid of points of C.
Find and record a proof that every continuous function defined on a compact (closed and bounded) interval is Riemann integrable.
Complete the details of the arguments of Example 1.3 on page 8 proving that f(x)= x^1/2 is integrable on [0,1] with integral 2/3.
Complete the details of the arguments of the two counter examples given on pages 10 and 11.
- In Example 1, what IS the integral of each function f_n and why is the integral what you say it is? Too, what IS the limit function, f, and why is f not Riemann integrable?
- In Example 2, why does the sequence {f_n} conveerge pointwise to 0? Too, why is this convergence NOT uniform?

Theorems and Definitions 1
1. Definitions
  - f is a function on a set A?
  - f is a relation on a set A?
  - the domain of a function f?
  - the range of a function f?
  - the union of sets A and B?
  - the union of sets A_n where n=1,2,... ?
  - the intersection of sets A and B?
  - the intersection of sets A_n where n=1,2,... ?
  - the complement of a set A?
  - If f and g are functions, what does it mean that f is an extension of g?
  - If f and g are functions, what does it mean that f is a restriction of g?
  - What does it mean for a function f to be continuous at a point a?
  - What does it mean for a function f to be differentiable at a point a?
  - What does it mean for a function f to be continuous on a set of points, A?
  - What does it mean for a function f to be differentiable on a set of points, A?
  - What is an equivalence relation on a set A and how is it usually expressed?
  - What does it mean for a relation to be symmetric?
  - What does it mean for a relation to be transitive?
  - What does it mean for a relation to be reflexive?
  - What does it mean for a set A to be an open set?
  - What does it mean for a set A to be a closed set?
  - What does it mean for a set A to be finite?
  - What does it mean for a set A to be infinite?
  - What does it mean for a set A to be countable?
  - What does it mean for a set A to be uncountable?
2. Theorem FTC x 2
3. Theorem DeMorgan's Laws
Did you check out Paul Erdös last week? No??, Click This!!