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(iii) Must the sequence (
x
) contain a convergent subsequence?
(iv) If the subsequence (
x
) converges to
c
, must the subsequence
c
?
(v) Why must
c
[
a
,
b
]?
(vi) What can be said about the sequences (
f
(
x
(
y
)
)) and (
f
(
y
)),
because of the continuity of
f
?
(vii) Usetheinequality
f
(
x
)
f
(
y
)
f
(
x
)
f
(
c
)
f
(
c
)
f
(
y
)
to obtain the contradiction we seek, by showing that the
right-hand side may be made less than the left-hand side.
44 In qns 38, 40 and 41 wefound continuous functions which wrenot
uniformly continuous. Each of these functions was, however, either
unbounded or else had unbounded slope. The function
f
: [0,
x
is unbounded and has
unbounded slope near the origin. Prove that this function is
uniformly continuous by appealing to qns 26 and 43 on [0, 1] and
by noting that, if either
x
or
y
1, then
x
y
x
y
.
45 If
f
:(
a
,
b
)
R is continuous and lim
)
R
defined by
f
(
x
)
f
(
x
) both
exist and are finite, prove that
f
is uniformly continuous by
constructing a function
g
which is continuous on [
a
,
b
] and for
which
g
(
x
)
f
(
x
) and lim
f
(
x
)on(
a
,
b
).
Extension of functions on
Q
to functions on
R
46
(i) Sketch the graph of
1
2
x
y
1
, for 0
x
1.
Use computer graphics if possible. You will need a large
rangefor
y
, say [
10, 10]. What is theequation of the
asymptote?
(ii) Is a real function defined by
f
(
x
)
1/(2
x
1) for all points
of thedomain [0, 1]
Q?
(iii) Is the function given in part (ii) continuous at each point of
thegiven domain?
(iv) As the curve is steep near the asymptote there is a possibility
that the function may not be uniformly continuous there.
Let
m
and
n
be positive integers with
m
n
.If
x
,
provethat
m
/2
f
(
x
)
n
/2.
