Graphics Reference
In-Depth Information
We can also express the uniform convergence of (
f
n
)to
f
by saying
that
(sup
f
n
(
x
)
f
(
x
)
:
x
A
)
0as
n
,
or by saying that, given
0, there exists an
N
such that
n
N
sup
f
n
(
x
)
f
(
x
)
:
x
A
.
15 Use qns 7 and 8 to prove that the sequences of functions (
f
) in qns
1 and 2 converge uniformly to their respective pointwise limit
functions
f
.
16 Useqns 12, 13 and 14 to provethat the sequences of functions (
f
)
in qns 3, 4 and 5 do not converge uniformly to their respective
pointwiselimit functions.
17 The convergence of a sequence of functions, uniform or otherwise,
may depend on the domain of those functions.
Examine the convergence of the sequence of functions given by
f
(
x
)
x
/
n
(i) on thedomain [
a
,
a
],
(ii) on thedomain R.
18 Examine the convergence of the sequence of functions in qn 4
(i) on thedomain [0,
a
], where 0
a
1, and
(ii) on thedomain [0, 1).
19 The convergence of a seemingly well-behaved sequence of functions
may fail to beuniform.
Examine the convergence of the sequence of functions given by
f
(
x
)
nx
/(1
n
x
) on thedomain R.
Draw thegraphs of
f
for
n
1, 2, 3, 5, 10.
This particular example is a useful corrective to a wrong impression
that might be gleaned from qns 1, 2, 3, 4 and 5. For in those questions
there was uniform convergence to constant functions, and non-uniform
convergence to non-constant functions. Question 19 shows that this
coincidence was accidental.
Uniform convergence and continuity
Having defined uniform convergence with the intention of giving a
condition that would make a sequence of continuous functions
