Section 5.1
#1)
By the definition of continuity,
f
is continuous at
c
if and only if lim
x
→
c
f
=
f
(
c
). By the Sequential
Criterion in Chapter 4, this is equivalent to the statement that if (
x
n
) is a sequence converging to
c
such that
x
n
=
c
, then (
f
(
x
n
)) converges to
f
(
c
). Clearly, this condition is satisfied if the Sequential
condition in 5.1.3 (which allows for ANY sequence converging to
c
) is satisfied. Conversely, let (
x
n
) be
any sequence converging to
c
. Either it is ultimately constant and equal to
c
, and there is nothing to
prove since
f
(
x
n
) will be ultimately constantly
f
(
c
), or the terms
x
n
k
not equal to
c
form an infinite
subsequece converging to
c
. Thus, if
f
is continuous at
c
,
f
(
x
n
k
) converges to
f
(
c
), and hence
f
(
x
n
)
converges to
f
(
c
), since every term not in
f
(
x
n
k
) is equal to
f
(
c
).
#3)
Let
>
0. If
x
0
∈
[
a, b
), then we can find
δ >
0 such that
x
0
+
δ < b
, and if
x
∈
[
a, b
],

x

x
0

< δ
,
then

f
(
x
)

f
(
x
0
)

<
. (Note that if
x
∈
[
a, c
],

x

x
0

< δ
, then automatically,
x
∈
[
a, b
).) Since
h
≡
f
on [
a, b
), then for
x
∈
[
a, c
], if

x

x
0

< δ
, then

h
(
x
)

h
(
x
0
)

=

f
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 JUNGE
 Topology, Continuity, Mathematical analysis, Metric space, Limit of a sequence

Click to edit the document details