MATH2400 ASSIGNMENT 3 SEMESTER 1 2023
Due at 1:00pm 22 May. Marks for each problem are shown. Total marks: 60
Submit your answers using the Blackboard assignment submission system.
(1) Let fn(x) =
✓
x 1
n
◆2
for x 2 [0, 1].
(a) (3 marks) Does the sequence {fn} converge pointwise in the set [0, 1]? If so, give
the limit function.
(b) (7 marks) Does the sequence {fn} converge uniformly on [0, 1]? Prove your asser-
tion.
(2) (10 marks) The Cartesian product X = X1 ⇥ X2 of two metric spaces (X1, d1) and
(X2, d2) can be made into a metric space (X, d) in many ways. For instance, show that
a metric d is defined by
d(x, y) = d1(x1, y1) + d2(x2, y2),
where x = (x1, x2), y = (y1, y2.)
(3) (10 marks) Fix an interval [a, b]. Let C[a, b] be the set of continuous functions from [a, b]
to R. For f, g 2 C[a, b], define a dot product and norm by
f · g :=
Z b
a
f(x)g(x) dx, kfk2 :=
p
f · f =
✓Z b
a
|f(x)|2 dx
◆1/2
(note the absolute value is actually not necessary). The dot product is clearly bilinear
and symmetric (you do not need to show this or that · defines a dot product). Show
that k·k2 is a norm on C[a, b].
(4) (10 marks) Consider the sequence of functions fn : [0, 1]! R given by
fn(x) =
8<:1 nx if 0  x  1/n,1 otherwise,
for n > 0, which converges pointwise to f(x) = 1 as n ! 1. Show that {fn}1n=1 does
not converge to f in the uniform norm, but it does converge using the norm defined in
Problem (3). (As a consequence, for infinite dimensional vector spaces, there are norms
that are not equivalent.)
(5) (10 marks) Let f : R2 ! R2 be defined by
f(x, y) :=
2 x+ 3y + y2, 3x 2y xy
Use directly the definition of the derivative to show that f is di↵erentiable at the origin
and compute f 0(0, 0).
Hint: If the derivative exists, it is in L(R2,R2), so it can be represented by a 2⇥2 matrix.
(6) Define f : R2 ! R by
f(x, y) =
8<:
xy
x2 + y2
if (x, y) 6= (0, 0),
0 if (x, y) = (0, 0).
(a) (4 marks) Show that
@f
@x
and
@f
@y
exists at all points (including the origin) and show
that these are not continuous functions.
(b) (3 marks) Is f continuous at the origin? Explain your answer.
(c) (3 marks) Does f have directional derivatives at the origin? Explain your answer.