ASSIGNMENT 1
There are two parts to this assignment. The first part is on WeBWorK — the link is available on the course
webpage. WeBWorK questions can be submitted directly online. They should be completed before you attempt
the second part of the assignment, which consists of the questions on this page. For these questions, you
are expected to provide full solutions with complete arguments and justifications. You will be graded on the
correctness, clarity and elegance of your solutions. Your answers must be typeset or very neatly written.
They must be stapled, with your name and student number at the top of each page.
1. Let
(
f (x) =
1
q
if x is a rational number
0
if x is irrational
p
q
in lowest terms with q > 0
.
Prove, using the δ − ε definition of limit, that f is discontinuous at all rational numbers.
Bonus: prove that f is continuous at all irrational numbers.
2. (a) Let f be a continuous function on the interval [a, b], with f (a) < f (b). Let L be a number strictly
between f (a) and f (b). In this question, you will prove the Intermediate Value Theorem; that is,
that there is some number c ∈ (a, b) such that f (c) = L.
Find the midpoint m of the interval [a, b]. If f (m) = L, terminate the process. If f (m) > L,
pick [a, m] to be your new interval. If f (m) < L, pick [m, b] to be your new interval. Continue
this process of picking the midpoint and then terminating the process or halving the interval ad
infinitum. If the process does not terminate, you end up with a set of nested intervals with left
endpoints l1 ≤ l2 ≤ l3 ≤ · · · and right endpoints · · · ≥ r3 ≥ r2 ≥ r1 . Prove that the left endpoints
and right endpoints converge to the same point c, and that f (c) = L. (You may use the Monotone
Convergence Theorem without proof.)
(b) Imagine a circle anywhere in the universe. Prove that, at any given time, there are two points on the
circle directly opposite each other with the same temperature.
3. Recall the function f that we defined in class to describe the spawning salmon population in a branch of
the Fraser River:
f : number of salmon in a given year → number of salmon in the following year.
Explain why it is reasonable to assume that f is differentiable on its domain. Your explanation should
be in one or two paragraphs with appropriate mathematical justification.