620-295 Real Analysis with applications
Problem Sheet 4
Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville VIC 3010 Australia
aram@unimelb.edu.au
Last updates: 9 September 2009
Sequences and series
- Define the following and give an example for each:
- (a)
metric space,
- (b)
complete (for a metric space),
- (c)
completion (of a metric space)
,
- Let
be a sequence in
. Show that if
is increasing and bounded then
converges.
- Let
be a sequence in
. Show that if
is bounded then
converges.
- Let
be a metric space and let
be a sequence in
. Show that if
converges then
is Cauchy.
- Give an example of a Cauchy sequence that
does not converge.
- Let
be a sequence in
. Show that if
converges then
.
- Let
be a sequence in
. Show that if
converges then
converges.
- Let
with
.
Prove that
.
- Prove that
.
- Prove that
.
- Prove that
.
- Prove that
.
Limits
- Define the following and give an example for each:
- (a)
continuous at
,
- (b)
,
- (c)
continuous,
- (d)
uniformly continuous,
- (d)
Lipschiz continuous,
- (e)
derivative at
,
- For each of the following, guess the limit and then prove the guess by using the
definition of limit:
- (a)
,
- (b)
,
- (c)
,
- (d)
,
- (e)
,
- (f)
,
- (g)
,
- (h)
,
- Evaluate the following limits:
- (a)
,
- (b)
,
- (c)
,
- (d)
,
- (e)
,
- (f)
,
- (g)
,
- (h)
,
- Evaluate the following limits:
- (a)
,
- (b)
,
- (c)
,
- (d)
,
- (e)
,
- (f)
.
Continuous functions
- Let
be such that
is continuous at
and
if
then
.
Show that if
then
is continuous at
.
- Let
be such that
is continuous at
and
if
then
.
Show that if
then
is continuous at
.
- Let
be an interval in
.
Let
be continous. Show that the function
given by
is continuous.
- Let
be an interval in
and let
and
be continous. Show that the function
given by
is continuous.
- (Thomae's function) Let
be given by
Show that
- (a)
If
then
is continuous at
, and
- (b)
If
then
is not continuous at
.
- Let
be an interval in
and let
and
be continous. Show that the function
given by
is continuous.
- Let
and let
be given by
Show that
is continuous.
- Is the function
given by
uniformly continuous?
- Is the function
given by
uniformly continuous?
- Is the function
given by
uniformly continuous?
- Is the function
given by
uniformly continuous?
- Is the function
given by
uniformly continuous?
- Is the function
given by
uniformly continuous?
- Is the function
given by
uniformly continuous?
- Let
be the function given by
. Show that
- (a)
is continuous,
- (b)
is uniformly continuous,
- (c)
,
- (d)
There does not exist
such that
,
- (e)
,
- (d)
There does not exist
such that
.
- Show that
the function
given by
has exactly 3 roots.
- Let
be an interval in
and let
be a continuous function.
Prove that
is an interval.
- Let
and
be intervals in
and let
be a surjective strictly monotonic continuous function.
Prove that the inverse function
exists and is strictly monotonic and continuous.
Differentiability
- Let
and let
be a function.
Let
and carefully define
. Prove that if
and
are functions then
,
whenever
and
exist.
- Let
be such that
is differentiable at
and
if
then
.
Show that
- (a)
if
then
is differentiable at
,
- (b)
if
then
,
- (c)
Show that
is infinitely differentiable.
- Let
be such that
is differentiable at
and
if
then
.
Show that
- (a)
if
then
is differentiable at
,
- (b)
if
then
,
- (c)
Show that
is infinitely differentiable.
- Let
be given by
Is
continuous at
?
Is
differentiable at
?
- Let
be given by
Is
continuous at
?
Is
differentiable at
?
- Let
be given by
Is
continuous at
?
Is
differentiable at
?
- Let
and assume that
is differentiable on
and continuous on
. Assume that the limit
exists. Prove that the right derivative
exists and that
.
- Let
and assume that
is differentiable at
. Show that
exists and equals
. Is the converse true?
- Prove that
.
- Prove that
.
Mean value theorem
- Use the mean value theorem to prove the following inequalities:
- (a)
for all
.
- (b)
for all
,
- (c)
for all
.
- Use the mean value theorem to show that if a function
is differentiable with
for all
then
is strictly increasing.
- Use the mean value theorem to show that if a function
is twice differentiable with
then
is strictly convex.
(
is strictly convex if
for all
and
.
Picard and Newton iteration
- Let
is given by
.
Estimate numerically the solution to
with
using Picard iteration.
- Let
is given by
.
Estimate numerically the solution to
with
using Newton iteration
(let
).
- Show that the equation
has a solution between 0 and 1. Transform the equation
to the form
for a suitable function
. Use Picard iteration to find the solution to 3 decimal places. (Try
).
- Show that the equation
has a solution between
and 2. Transform the equation
to the form
for a suitable function
. Use Picard iteration to find the solution to 3 decimal places. (Try
).
Topology
- Define the following and give an example for each:
- (a)
metric space,
- (b)
limit of
as
approaches
,
- (c)
limit of
as
,
- (j)
continuous at
,
- (c)
continuous,
- (d)
uniformly continuous,
- (e)
Lipschitz,
- (f)
-ball,
- Define the following and give an example for each:
- (a)
topology,
- (b)
topological space,
- (c)
open set,
- (d)
closed set,
- (e)
interior,
- (f)
closure,
- (g)
interior point,
- (h)
close point,
- (i)
neighborhood,
- (j)
fundamental system of neighborhoods,
- (k)
continuous at
,
- (l)
continuous,
- Define the following and give an example for each:
- (a)
topological space,
- (b)
Hausdorff,
- (b)
fundamental system of neighborhoods,
- (b)
basis,
- (c)
connected set,
- (d)
compact set,
- Prove that
and
is a basis of
if and only if
satisfies:
if
then
is a fundamental system
of neighborhoods of
.
- Let
and
be metric spaces. Define the topology on
and
.
Prove that
is continuous as a function between metric spaces
if and only if
is continuous as a function between topological spaces.
- Define the following and give an example for each:
- (b)
filter,
- (c)
finer,
- (b)
filter base,
- (b)
neighborhood filter,
- (d)
limit of
as
approaches
,
- (b)
Fréchet filter,
- (d)
limit of
as
.
- Define the following and give an example for each:
- (c)
ultrafilter,
- (d)
quasicompact,
- (d)
Hausdorff,
- (d)
compact,
- Let
be a Hausdorff topological space and let
be a compact subset of
. Show that
is closed.
- Let
be a metric space. Show that
is Hausdorff and has a countable basis.
- Let
be a metric space and let
be a compact subset of
. Show that
is closed and bounded.
- Let
be a metric space and let
be a subset of
. Show that
is compact if and only if every infinite subset of
has a limit point in
. (What is the definition of limit point???)
- Let
be a subset of
. Show that
is compact if and only if
is closed and bounded.
- Let
and
be topological spaces and let
be a continuous function. Show that
if
is connected then
is connected.
- Let
. Show that
is connected if and only if the
set satisfies
if
and
and
then
.
- (Intermediate Value Theorem) Let
be a continuous function. Show that
if
and
then there exists
such that
.
- Let
and
be topological spaces and let
be a continuous function. Show that
if
is compact then
is compact.
- Let
be a closed bounded subset of
and let
be a continuous function.
- (a)
is a bounded function,
- (b)
attains its maximum and minimum on
,
- (a)
is uniformly continuous.