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converge to a continuous limit, we need to see whether the condition
we have given is suMcient to guarantee this in every case.
20 We suppose that the sequence of functions (
f
) converges uniformly
to thefunction
f
and that each of the functions
f
is continuous on
its domain
A
. We seek to prove that
f
is continuous on
A
or, in
other words, that, for each
a
A
, lim
f
(
x
)
f
(
a
).
Or, again, given
0, there exists a
such that
x
a
f
(
x
)
f
(
a
)
.
Webreak down
f
(
x
)
f
(
a
) into manageable pieces.
f
(
x
)
f
(
a
)
f
(
x
)
f
(
x
)
f
(
x
)
f
(
a
)
f
(
a
)
f
(
a
)
f
(
x
)
f
(
a
)
f
(
x
)
f
f
f
f
f
(
a
)
so
(
x
)
(
x
)
(
a
)
(
a
)
.
For what reason must both
f
(
x
)
f
(
x
)
and
f
(
a
)
f
(
a
)
belss
than
for suMciently large
n
?
Keeping to one of these suMciently large
n
s, for what reason is it
possibleto find a
such that
x
a
f
(
x
)
f
(
a
)
?
Now complete the proof.
21 We have proved that the uniform limit of a sequence of continuous
functions is continuous.
By considering qns 17(ii) and 19, show that the converse of this
theorem is false: namely that a non-uniform limit of a sequence of
continuous functions may also becontinuous.
22 If (
f
) is a sequence of continuous functions which converges
uniformly to thefunction
f
, and
a
is a point in thedomain of thse
functions, justify each step of the following argument.
lim
lim
f
(
x
)
lim
f
(
a
)
f
(
a
)
lim
f
(
x
)
lim
lim
f
(
x
).
23 Illustrate the dependence of the argument in qn 22 on uniform
convergence by showing how it would fail for the sequence of
functions in qn 3 if wewreto take
a
0.
