MATH 142B Homework 3
Due Monday 11:00pm, May 8, 2023 (page 1 of 1)
1. Let f : [0, 1]→ R be defined by
f(x) =
{
x, x ∈ Q
0 x /∈ Q.
(a) Calculate the upper and lower Darboux integrals of f .
(b) Is f integrable? Be sure to justify your answer.
2. Let f : [a, b] → R be a bounded function. Suppose that there exist sequences (Ln)n∈N and
(Un)n∈N of upper and lower Darboux sums for f such that limn→∞(Un − Ln) = 0. Prove that
f is integrable and
∫ b
a
f = lim
n→∞Ln = limn→∞Un.
3. Prove that if f : [a, b]→ R is integrable, then f is integrable on every subinterval [c, d] ⊆ [a, b]
4. Prove that a decreasing function f : [a, b]→ R is integrable.
5. Let f : [a, b] → R be integrable and suppose g is a function on [a, b] which coincides with f
except at finitely many points in [a, b]. Prove that g is integrable on [a, b] and that
∫ b
a
g =
∫ b
a
f .
6. Exhibit an example of a function f : [0, 1] → R that is not integrable but for which |f | is
integrable.
7. Let f : [a, b] → R be a bounded function so that there exists B > 0 for which |f(x)| ≤ B for
all x ∈ [a, b].
(a) Prove that
U(f2, P )− L(f2, P ) ≤ 2B(U(f, P )− L(f, P ))
for all partitions P of [a, b].
(b) Prove that if f : [a, b]→ R is integrable, then f2 is also integrable.
(c) Exhibit a function f : [0, 1]→ R which is not integrable but for which f2 is integrable.
8. Let f : [a, b]→ R and g : [a, b]→ R be integrable functions.
(a) Prove that fg is integrable.
(b) Prove that max(f, g) and min(f, g) are integrable.
9. Suppose f : [a, b] → R and g : [a, b] → R are continuous functions such that
∫ b
a
f =
∫ b
a
g.
Prove that there exists x ∈ (a, b) at which f(x) = g(x).
10. (a) Prove that if f : [a, b] → R and g : [a, b] → R are continuous functions with g(t) ≥ 0 for
all t ∈ [a, b], then there exists x ∈ (a, b) such that∫ b
a
f(t)g(t) dt = f(x)
∫ b
a
g(t) dt.
(b) Show that the Intermediate Value Theorem for Integrals is a special case of part (a).
(c) Does the conclusion in part (a) hold if [a, b] = [−1, 1] and f(t) = g(t) = t for all t ∈ [a, b]?
Be sure to justify your answer.