MATH2320/6110: Analysis 1
2024 Add-on Assignment 2 — Due Tuesday 28th May
This is really just a suggested date for you to submit.
Please send an email to Griff: griffith.ware@anu.edu.au
if you need a later due date1.
Most of these questions are exercises from Sections 1A and 1B in Measure, Integration & Real Analysis by
Sheldon Axler, available free online.
(1) We will write L
∫ b
a f for the lower Riemann integral and U
∫ b
a f for the upper Riemann integral, of a
bounded function f : [a, b] −→ R.
(a) Suppose that f : [a, b] −→ R and g : [a, b] −→ R are bounded functions. Prove that
L
∫ b
a
f + L
∫ b
a
g ≤ L
∫ b
a
(f + g)
and
U
∫ b
a
f + U
∫ b
a
g ≥ U
∫ b
a
(f + g).
(b) Give an example of functions f and g to show that the two inequalities in part (a) can be strict.
(2) Suppose f : [a, b] −→ R and g : [a, b] −→ R are Riemann-Darboux integrable functions.
Prove that f + g is Riemann-Darboux integrable on [a, b], and∫ b
a
(f + g) =
∫ b
a
f +
∫ b
a
g.
[Hint: use the result of Q1(a).]
(3) Suppose f : [a, b] −→ R is Riemann-Darboux integrable.
Prove that if c, d ∈ R and a ≤ c < d ≤ b, then f |[c,d] is Riemann-Darboux integrable on [c, d].
[Hint: use the property that a function g is Riemann-Darboux integrable if and only if for all ε > 0
there exists a partition P of the domain of g such that U(g, P )− L(g, P ) < ε.]
(4) Suppose f : [a, b] −→ R is Riemann-Darboux integrable, and c ∈ (a, b). Prove that f is Riemann-
Darboux integrable on [a, b] if and only if f |[a,c] is Riemann-Darboux integrable on [a, c] and f |[c,b] is
Riemann-Darboux integrable on [c, b]. Furthermore, prove that if these conditions hold then∫ b
a
f =
∫ c
a
f +
∫ b
c
f.
[Hint: use the results of Q2 and Q3.]
(5) Suppose f : [a, b] −→ R is an increasing function, meaning that c, d ∈ [a, b] with c < d implies
f(c) ≤ f(d). Prove that f is Riemann-Darboux integrable on [a, b].
[Hint: you can give a very short answer to this by quoting results from an earlier main-course assign-
ment and results mentioned in add-on lectures.]
[Q6 is on the next page.]
1There will be a limit based on when I need to have things finalised at a departmental level for final grade submission.
1
(6) In the main-course Assignment 4 Question 1, the definition of the Riemann integral was given (as
opposed to the Riemann-Darboux integral definition we have been using in the add-on). Furthermore,
a particularly useful property was shown in part (iv) of Q1 in that assignment. The sequence of results
from that assignment shows that the property in Q1(iv) holds for any bounded function f . Also note
that the I defined in that part is actually I = L
∫ b
a f , using the notation from the add-on.
(a) Without proving anything, modify the property from Q1(iv) appropriately to write down the
corresponding statement for U
∫ b
a f . This corresponding statement is proved in a very similar
‘mirror image’ manner. You may use both the statement for U
∫ b
a f , and the one for L
∫ b
a f from
A4 Q1(iv), in the following question, without proof.
(b) Suppose f : [a, b] −→ R is bounded. Prove that f is Riemann-Darboux integrable if and only f
is Riemann integrable, and that in such cases the value of the two integrals are the same. (Hence
Riemann-Darboux and Riemann integrability are equivalent notions.)