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x
at
x
establish the sequence definition in the case of
f
(
x
)
3. Can
we show that this neighbourhood definition of continuity is equivalent
to the sequence definition that we have used in the first half of this
chapter?
67 Suppose that the neighbourhood definition of continuity holds for a
function
f
and a point
x
a
in thedomain of thefunction. Lt
(
a
a
be a sequence in the domain of the function. Can we show
that the sequence (
f
(
a
)
))
f
(
a
)?
f
(
a
) according to the
standard definition of convergence, qn 3.60.
(ii) Use the neighbourhood definition of continuity to find a
neighbourhood of
a
, such that
f
(
a
(i) Writedown what is meant by (
f
(
a
))
)
f
(
a
)
, when
a
is in
thenighbourhood.
(iii) Use the convergence of (
a
) to show that terms of the
sequence (
a
) eventually lie in the neighbourhood you found
in part (ii).
Question 67 establishes that neighbourhood continuity at a point
implies sequence continuity at that point. Can we show that sequence
continuity implies neighbourhood continuity? The sequence definition
of continuity considers a collection of sets of points
a
, clustering
about
a
, in thedomain of thefunction
f
. Although all the sequences
converge to
a
, the definition seems to leave in doubt the question as to
whether, if
f
is sequentially continuous at
a
, given
0, there must be a
-neighbourhood of
a
such that,
whenever x
belongs to that neighbour-
hood,
f
(
x
)isinan
-neighbourhood of
f
(
a
).
To clarify this, we suppose that sequence continuity holds and
neighbourhood continuity fails, for some
, so that every
neighbourhood of
a
contains roguepoints
x
, for which
f
(
x
)
f
(
a
)
,
and weobtain a contradiction.
68 Supposethat
f
is a real function which is sequentially continuous at
a
, and that, for some
0, there is no
-neighbourhood of
a
such
that, for all
x
in thenighbourhood,
f
(
x
)
f
(
a
)
.
(i) Why must there be a value of
x
within thenighbourhood
x
a
1/
n
x
a
1/
n
such that
f
(
x
)
f
(
a
)
?
(ii) If thevalueof
x
in (i) is called
a
, how do you know that
a
) is a null sequence?
(iii) With thenotation of (i) and (ii), how do you know that the
sequence (
f
(
a
(
a
)) does
not
tend to
f
(
a
)? This contradicts our
starting point.
The contradiction obtained in qn 68 leads to the conclusion that, if
