Graphics Reference
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on a number line. They are, in fact, identical. Such a set is called a
2-neighbourhood of the point 1. The neighbourhood has centre 1
and radius 2. Comparethis claim with qn 3.61.
57 Provethat, if
x
2
0.2, then
x
4
1.
Illustratethis rsult on a graph of thefunction
x
x
.
58 Provethat, if
x
3
0.1, then
x
9
1.
59 Provethat, if
x
2
1/3, then
1/
x
1/2
1/10.
60 Provethat, if
x
2
0.02, then
x
2
0.01.
61 By considering
x
2 in qn 57,
x
3 in qn 58,
x
2.4 in qn 59
and
x
1.975 in qn 60, show that noneof theimplications that
you have established in qns 57
—
60 may be reversed.
Each of these implications takes theform:
if
x
a
, then
f
(
x
)
f
(
a
)
.
Expressed rather crudely, each says that if
x
is near to
a
then
f
(
x
)is
near to
f
(
a
).
If wewant to pinpoint continuity this way, it mattrs just
how
near is
near
. We can investigate the kind of connection between
and
that is needed for continuity by examining a point of
discontinuity.
62 Let
f
(
x
)
2.
If
x
2
2, can you find an
such that
f
(
x
)
f
(2)
?
If
[
x
], the integer function, and let
a
x
2
1, can you find an
such that
f
(
x
)
f
(2)
?
If
x
2
, can you find an
such that
f
(
x
)
f
(2)
?
x
f
(
x
)
f
(2)
If
2
, can you find an
such that
?
Thefact that wecan find an
every time in qn 62 tells us nothing
whatever about the continuity of
f
.
If we turn the question round, it begins to bite.
63 Let
f
(
x
)
[
x
] as before.
Can you find a
such that, if
x
2
,
then
f
(
x
)
f
(2)
2?
Can you find a
such that, if
x
2
,
then
f
(
x
)
f
(2)
1?
Can you find a
such that, if
x
2
,
then
f
(
x
)
f
(2)
?
Can you find a
such that, if
x
2
,
then
f
(
x
)
f
(2)
?
