11. Is there a function
f
which is continuous on
R
and
f
(1
/n
) = (

1)
n
for all positive
integer
n
?
12. Is there a function
f
which is continuous on
R
and
f
(1
/n
) = (

1)
n
/n
for all positive
integer
n
?
13. We deﬁne the positive part of a number
x
as
x
+
=
(
x
if
x
≥
0
0
if
x <
0
(a) Show that the function
x
→
x
+
is continuous everywhere.
(b) Assume that
g
is continuous everywhere. Show that
g
+
(
x
) (the positive part of
g
(
x
)) is also continuous everywhere.
(c) Express the maximum of
x
and
y
using the positive part function. Use that
to show that if
f
and
g
are continuous functions then the function
h
(
x
) =
max(
f
(
x
)
,g
(
x
)) is also continuous.
14. Show that the equation
x
3
= sin(
x
2
) + cos(4
x
+ 3) has at least one solution.
15. Show that the function
x

tan
x
has inﬁnitely many zeros.
16. The function
f
is continuous on [0
,
∞
) and lim
x
→∞
f
(
x
) = 12. Show that
f
is bounded
on [0
,
∞
).
Ask for help if you think you need it
If you are having trouble with certain type of problems or concepts then you should ask for
help. Come to one of my or Jo’s oﬃce hours and ask questions! If you think you may have
trouble solving problems with a time limit then collect a couple (say ﬁve) problems similar
to homework problems and try solving them in 90 minutes (with the solutions written up
neatly). Remember that it is almost as important that you can present your solutions clearly
as it is to actually ﬁnd those solutions.
GOOD LUCK!
3
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 Fall '08
 Staff
 Math, Calculus, Continuous function, Value Theorem, basic limit laws

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