ASSIGNMENT 1·8
There are two parts to this assignment. The first part is on WeBWorK — the link is available on the course
webpage. The second part consists of the questions on this page. 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. Find c such that the following function is continuous:
x+c
if x < c
.
f (x) =
(x − 1)(x − 1 − c) if x ≥ c
2. Let f (x) =
x2
0
if x is rational
.
if x is not rational
(a) Prove, using the limit definition of derivative, that f is differentiable at 0.
(b) Prove that f is not continuous at 1. (The same technique may be used to show that f is not continuous
at any nonzero number.)
3. Let d(t) be the shortest distance between Earth and the nearest other planet as a function of time t
(where t is in years and t = 0 is noon on January 1, 1000 A.D., say). Is d continuous? Why or why not?