SCIENCE ONE: MATHEMATICS ASSIGNMENT 8
There are two parts to this assignment. The first part is called Assignment 8 (online); you can find it at
www.mathxl.com. 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 primarily 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
0
if x = 15 , 25 , 35 or
otherwise
4
5
. Prove that f is integrable on [0, 1], and find its integral.
2. Recall Thomae’s function from your first assignment:
(
1
if x is rational and of the form pq where the fraction is in lowest terms and q > 0
q
.
f (x) =
0 if x is not rational
You proved that f is continuous at irrational numbers and discontinuous at rational numbers. Prove that
f is integrable on [0, 1], and find its integral. (Hint: find its lower Riemann sum; then show that there
are only finitely many points above any line y = ε, where ε > 0.)
3. In class, we proved the Mean Value Theorem for integrals: if f is continuous on [a, b], then there exists
some point c in [a, b] such that
Z b
f (c)(b − a) =
f (x)dx.
a
Prove that c need never be a or b. In other words, prove that we can replace the phrase there exists some
point c in [a, b] with the phrase there exists some point c in (a, b).