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With our success in finding
s in qn 62 and our failureto find
sin
qn 63, it looks as if wecan pin down continuity at
x
a
by saying
f
is continuous at
x
a
,
provided that, whatever positive
we choose,
we can find a
such that
x
a
implies
f
(
x
)
f
(
a
)
.
We first of all see whether this holds in cases that we know are
continuous as in qns 64(i) and (ii).
64
(i) Find a
such that
x
3
implies
x
9
0.5.
(ii) Find a
1/6.
(iii) Find a
such that
x
3
implies
x
3
0.01.
such that
x
3
implies
1/
x
1/3
Notice that in each case here, when a satisfactory
has been found,
any lesser (but still positive)
will also serve.
65
(i) If
x
3
1, provethat
x
3
/(
2
3).
(ii) Given
0, find a
such that
x
3
x
3
.
Question 65(ii) establishes that the neighbourhood definition of
continuity holds for thefunction
x
x
at thepoint
x
3. The
next question uses this to show that the sequential definition of
continuity holds for this function at this point. That is, the
neighbourhood definition of continuity implies the sequential
definition.
66 We seek to prove that for any sequence of positive terms (
a
)
3,
wehave(
a
)
3.
(i) Writedown what is meant by (
a
)
3, according to the
standard definition.
(ii) Write down what is guaranteed by the neighbourhood
definition of continuity of
x
x
at
x
3.
(iii) Now usethestandard definition of (
a
)
3, and (ii), to
establish (i).
So we know that the 'neighbourhood definition of continuity' fails in a
case(qn 63) whreweknow thefunction is not continuous and wehave
found the neighbourhood definition of continuity holds in two cases
(qns 64(i) and 64(ii)) where we know the function is continuous. And in
qn 66 we have used the neighbourhood definition of continuity to
