18.100B/C: Fall 2008
Homework 6
Available
Wednesday, October 15
Due
Wednesday, October 22
1.
(10pts) Problem 1, page 98 in Rudin
2.
(10pts) Problem 3, page 98 in Rudin
3.
(10 pts) Let
(
X, d
)
be a metric space. Fix
x
0
∈
X
and a continuous function
g
:
R
→
R
.
Show that the function
X
→
R
deﬁned by
x
±→
g
(
d
(
x, x
0
))
is continuous.
4.
Let
(
S, d
S
)
be a set equipped with the discrete metric (i.e.
d
S
(
t, r
) = 1
for
t
²
=
r
).
(a)
(5pts) Show that any map
f
:
S
→
X
into another metric space
X
is continuous; using
the deﬁnition of continuity by sequences.
(b)
(5pts) Show that any map
f
:
S
→
X
into another metric space
X
is continuous; using
the deﬁnition of continuity by
±
 and
δ
balls.
(c)
(10pts) Which maps
f
:
R
→
S
are continuous? (Give an easy characterization and
prove it.)
5.
Consider the function
h
:
Q
→
R
given by
h
(
x
) =
(
0
;
x
2
<
2
,
1
;
x
2
>
2
.
(a)
(5pts) Is
h
continuous?
(a)
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 Fall '10
 Prof.KatrinWehrheim
 Topology, Continuous function, Metric space

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