ASSIGNMENT 2
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 justifications. You will be graded on the mathematical, logical and grammatical coherence and
elegance of your solutions. Your solutions must be typed, with your name and student number at the top of
the first page. If your solutions are on multiple pages, the pages must be stapled together.
Your written assignment must be handed in before the start of your recitation on Friday, September
19. The online assignment will close at 9:00 a.m. on Friday, September 19.
1. Find all values of a and b that make the function
 2
x −4

 x−2
f (x) =
ax2 − bx − 2

 4x − a + b
if x < 2
if 2 ≤ x < 3
if x ≥ 3
continuous everywhere.
2. Let
(
f (x) =
1
q
0
if x = pq is a rational number in lowest terms with q > 0
if x is irrational
(a) Prove using the definition of limit that f is not continuous at any rational number c.
(b) [Bonus] Is f continuous at any irrational numbers?
3. Consider a circle anywhere in the universe. Prove that at any time there are two antipodal points on the
circle that have the same temperature. (Hint: consider the difference in temperature between any two
antipodal points, and use the Intermediate Value Theorem.)