Graphics Reference
In-Depth Information
6 A real function
f
is continuous on (0, 1) and takes positive values.
Which one of the following statements must be true?
(i)
f
is bounded on (0, 1);
(ii)
f
(
x
) tends to a non-negative limit as
x
0
;
(iii)
f
attains a minimum though not necessarily a maximum,
value;
(iv)
the Riemann integral
f
exists;
(v) noneof thefour abov.
7 A real function
f
is continuous on the open interval (0, 1).
(i) If
f
is uniformly continuous, must
f
bebounded?
(ii) If
f
is bounded, must
f
beuniformly continuous?
(iii) May
f
be neither bounded, nor uniformly continuous?
8 A differentiable function
f
: R
R is strictly monotonic increasing.
Must its derivative be positive?
Problems for corporate or individual investigation
9 In many topics Archimedean order is claimed by saying that, if two
positive numbers are given, then some positive integer multiple of
the first exceeds the second. Is this claim equivalent to the Property
of Archimedean order which we have used in chapter 3?
10
On the set of formal power series
a
x
for some
integer
m
, and
a
R
, can you construct
,
and
so that the
set is an ordered field?
You will need to take
x
x
1
1/
x
1/
x
. Is the field then
Archimedean ordered?
Must Cauchy sequences converge?
11 Find the cluster points of the bounded sets
(i)
. Is there a subsequence of
(sin
n
) which tends to 0? If so, can you find one?
sin
n
n
N
, (ii)
n
2
[
n
2]
n
N
12 Does (
sin
n
) havea limit?
