ASSIGNMENT 4: MATH2320/6110
DUE TIME: 5:00PM, 12 MAY 2024
Question 1. (2 + 2 + 2 + 9 + 7 = 22 points) This question is designed to show
continuous functions on a closed bounded interval [a, b] are Riemann integrable. Recall that
a partition P of [a, b] is a finite sequence x0, x1, · · · , xn such that
a = x0 < x1 < · · · < xn = b.
Let ∆xi := xi − xi−1. Then
|P | := max
1≤i≤n
∆xi
is called the meshsize of this partition. Another partition P ′ of [a, b] is called a refinement
of the partition P , if P ′ contains all the points of P and possibly some other points as well.
A function f : [a, b]→ R is called Riemann integrable on [a, b] if there is a number I such
that for any ε > 0 there is a number δ > 0 such that for any partition P = {a = x0 < x1 <
· · · < xn−1 < xn = b} of [a, b] with |P | < δ and any choice of ξi ∈ [xi−1, xi] there holds∣∣∣∣∣
n∑
i=1
f(ξi)∆xi − I
∣∣∣∣∣ < ε.
The number I is then called the Riemann integral of f on [a, b] and is denoted as
∫ b
a
f(x)dx.
Show that a continuous function f : [a, b] → R is Riemann integrable by completing the
following procedure:
(i) Show that f is a bounded function on [a, b], i.e. there is a number B such that
|f(x)| ≤ B for all x ∈ [a, b].
(ii) Given a partition P = {a = x0 < x1 < · · · < xn−1 < xn = b} of [a, b], let
mi = inf{f(x) : x ∈ [xi−1, xi]} for each i = 1, · · · , n and define
L(f, P ) :=
n∑
i=1
mi∆xi.
Show that L(f, P ) ≤ B(b− a).
(iii) Let P ′ denote a refined partition of P of [a, b]. Show that L(f, P ) ≤ L(f, P ′).
(iv) Define
I := sup {L(f, P ) : P is a partition of [a, b]} .
According to (ii), I is a finite number. Show that for any ε > 0 there is a δ > 0
such that for any partition P of [a, b] with |P | < δ there holds
0 ≤ I − L(f, P ) < ε.
(v) Show that if f is continuous on [a, b], then f is Riemann integrable on [a, b] and∫ b
a
f(x)dx = I.
Question 2. (5 + 5 + 2 = 12 points) Let (X, d) and (Y, ρ) be metric spaces and let
X × Y be the metric space endowed with the metric
D((x1, y1), (x2, y2)) := d(x1, x2) + ρ(y1, y2), ∀(x1, y1), (x2, y2) ∈ X × Y.
Consider a function f : X → Y . The graph of f is the subset of X × Y defined by
G := {(x, y) ∈ X × Y : y = f(x)}.
1
2(a) Show that if f is continuous, then G is a closed subset of X × Y ,
(b) Assume that (Y, ρ) is compact. Show that if G is closed, then f is continuous.
(c) Give an example to show that if (Y, ρ) is not compact, then f : X → Y may be
discontinuous and still have closed graph.
Question 3. (4 + 12 = 16 points) Let (X, d) and (Y, ρ) be two metric spaces. The
following are some senses in which (X, d) and (Y, ρ) can be regarded as “similar”.
(A) (X, d) and (Y, ρ) are isometric if there is a bijection f : X → Y that preserves
distances, in the sense that ρ(f(x), f(z)) = d(x, z) for all x, z ∈ X.
(B) (X, d) and (Y, ρ) are comparable if there is a bijection f : X → Y such that f and f−1
are Lipschitz continuous functions. Here, a function f : X → Y is called Lipschitz
if there is a constant L such that ρ(f(x1), f(x2)) ≤ Ld(x1, x2) for all x1, x2 ∈ X.
(C) (X, d) and (Y, ρ) are uniformly equivalent if there is a bijection f : X → Y such that
f and f−1 are uniformly continuous.
(D) (X, d) and (Y, ρ) are homeomorphic if there is a bijection f : X → Y such that f
and f−1 are continuous.
Please answer the following questions to understand these concepts:
(i) Show the implications (A) =⇒ (B) =⇒ (C) =⇒ (D).
(ii) For each implication in (i) give an example of two metric spaces showing the impli-
cation is not reversible.