Write your name clearly on all sheets of paper you will turn in and possibly number
them.
Write clearly and large enough to be easily readable.
Your proofs must be
complete
and clearly written. All scalars appearing in the exercises are supposed to
be real if not differently stated.
Each question is worth the number of points indicated. Fifty points will grant you an
A.
1 Let
f
(
x
) be a bounded measurable function from
R
n
to
R
such that
lim sup

f
(
x
)
k
x
k
α
=
C <
∞
where
k
x
k
=
p
x
2
1
+
· · ·
+
x
2
n
.
a) (7 pts) For which values of
α
can you say that
f
∈
L
1
(
R
n
).
b) (5 pts) Is your condition on
α
necessary?
2 Given
f
∈
L
1
(
R
) define
F
(
x
) =
R
x
∞
f
(
y
)
dy
.
a) (7 pts) Show that
F
(
x
) is a continuous function. Is
F
(
x
) differentiable?
b) (10 pts) Show that for every
g
∈
C
1
0
(
R
) we have:
Z
R
F
(
x
)
g
0
(
x
)
dx
=

Z
R
f
(
x
)
g
(
x
)
dx.
Remember that
C
1
0
(
R
) is the set of all function in
C
1
(
R
) that are 0 outside a
compact set. Moreover
C
1
(
R
) is the set of all function that are continuous and
admit a continuous derivative.
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 Spring '08
 Staff
 Topology, Scalar, pts, Continuous function, R R, Dominated convergence theorem

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