±
±
18.100A
Introduction
to
Analysis
–
Practice
Final
3
hours
Directions.
You can use the textbook, but no other material.
To justify the arguments,
quote
theorems
, by name or number, verifying their hypotheses
where necessary.
Work problems totalling 100.
You can include just parts of some
problems.
Problems will be graded in their numerical order, stopping at 100 or as close
above it as possible, the total scaled back to 100 if it goes over.
∞
1
1.
(15: 5, 10)
Let
f
(
x
) =
.
x
2
+
n
2
1
a)
Prove that
f
(
x
) is deFned and continuous on (
−∞
,
∞
). (Do not use part (b).)
b) Prove
f
(
x
) is diﬀerentiable on (
−∞
,
∞
).
(Hint: Use intervals of the form [
−
a, a
].)
2.
(15: 10,5)
a)
Let
S
be a compact subset of the open Frst quadrant
Q
=
{
(
x, y
) :
x >
0
, y >
0
}
.
Prove there is a point
x
in
S
such that the ray connecting it to the origin has maximum
slope (i.e., no other point in
S
gives a ray with bigger slope).
b)
Does (a) continue to be true if
S
is only assumed to be closed, not compact? If yes,
show how to modify the proof in (a) to prove it; if no, give a counterexample.
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 Fall '12
 ArthurMattuck
 Topology, Metric space, Compact space, Cauchy sequence, R. Prove

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