# ASSIGNMENT 3

```ASSIGNMENT 3
There are three parts to this assignment. Part A is to be completed online before 10:00 a.m. on Monday,
October 4. Part B, which should be typeset (preferably with LaTeX), is to be submitted to the assignment box
by the same time. Part C is a question to be completed by your Small Group — only one submission per
group, with the names of all participating members at the top.
Include your full name and student ID, along with the subject and assignment number, at the top of the front
page.
Part A [20 marks]
This part of the assignment can be found online, labelled Assignment 3, at www.mathxl.com. Questions
about the MathXL program itself should be sent to Eric; in fact, questions sent through the site will be
directed to him. To ask me questions, email me directly or come to my office.
Part B [20 marks]
1. Determine if — and if so, where — the following functions are discontinuous.
2
x if x is rational
(a) f (x) =
0
otherwise
(
1
if x is rational and of the form pq where p, q ∈ Z, gcd(p, q) = 1 and q &gt; 0
q
(b) f (x) =
0 otherwise
2. Let f be a continuous function with domain and range [a, b]. Prove that there exists a point c ∈ [a, b]
such that f (c) = c.
Part C [5 marks]
Imagine a circle anywhere in the universe. (For example, draw a circle on a sheet of paper, or imagine the
equator is a circle.) Prove that there are two points directly opposite each other on the circle with the same
temperature.
```