McGill University
Department of Mathematics and Statistics
MATH 243 Analysis 2, Winter 2024
Assignment 3
You should carefully work out all problems. However, you only have to hand in solutions
to problems 1 and 4.
This assignment is due Thursday, February 1, at 8:00pm. Late assignments will
not be accepted!
1. Let 0 < c < 1 and let
f : [0, 1]→ R, f(x) :=
{
1 if x = c
0 if x 6= c
Prove directly from the definition of Riemann integrability that f is Riemann integrable on [0, 1]
and that
∫ 1
0
f = 0.
Hint: Let δ > 0 and let P be a partition of [0, 1] of mesh less than δ. Show that 0 ≤ U(f, P ) < 2δ
by distinguishing the cases that c is a partition point of P , and that c is not a partition point
of P .
2. Let f : [0, 1]→ R be defined by
f(x) :=

−1 if x = 0
0 if 0 < x < 1
1 if x = 1
Prove directly from the definition of Riemann integrability that f is Riemann integrable on [a, b]
and that
∫ b
a
f = 0.
3. Let c ∈ [a, b] and let
f : [a, b]→ R, f(x) :=
{
0 if a ≤ x < c
1 if c ≤ x ≤ b
Prove directly from the definition of Riemann integrability that f is Riemann integrable on [a, b]
and determine
∫ b
a
f .
4. Let f : [a, b] → R be Riemann integrable and non-negative (i.e. f(x) ≥ 0 for all x ∈ [a, b]).
Prove that
∫ b
a
f ≥ 0.
Hint: Prove first that for any partition P of [a, b] it holds that U(f, P ) ≥ 0.