Modern Analysis, Homework 4
1. Suppose that
f
:
R
2
→
R
2
is defined by
f
(
x, y
) = (
x
2
+
xy, y
). Prove
directly from the
definition
that
f
is differentiable at each point of
R
2
.
2. Rudin, Ch 9 # 14: Define
f
(0
,
0) = 0 and
f
(
x, y
) =
x
3
x
2
+
y
2
if (
x, y
)
6
= (0
,
0)
.
(a) Prove that
D
1
f
and
D
2
f
are bounded functions in
R
2
. (Hence
f
is continuous.)
(b) Let
~u
be any unit vector in
R
2
. Show that the directional derivative (
D
~u
f
)(0
,
0)
exists, and that is absolute value is at most 1.
(c) Let
γ
be a differentiable mapping of
R
into
R
2
(in other words,
γ
is a differentiable
curve in
R
2
), with
γ
(0) = (0
,
0) and

γ
0
(0)

>
0. Put
g
(
t
) =
f
(
γ
(
t
)) and prove
that
g
is differentiable for every
t
∈
R
. If
γ
∈
C
1
(
R
,
R
2
), prove that
g
∈
C
1
(
R
,
R
).
(
Note:
Rudin does not assume the following, but you may: for all
t
∈
R
at which
γ
(
t
) = (0
,
0), we have

γ
0
(
t
)

>
0. I suspect, that by looking at a function such as
the monster in 4a, that this assumption really is necessary.)
(d) In spite of this, prove that
f
is not differentiable at (0
,
0). (
Hint:
Writing
~u
in
components:
~u
=
∑
u
i
e
i
, the formula
(
D
~u
f
)(
x
) =
X
(
D
i
f
)(
x
)
u
i
does not hold.)
3. Rudin, Ch 9 #14 (mostly): Define
f
(0
,
0) = 0, and put
f
(
x, y
) =
x
2
+
y
2

2
x
2
y

4
x
6
y
2
(
x
4
+
y
2
)
2
if (
x, y
)
6
= (0
,
0).
(a) Prove that for all (
x, y
)
∈
R
2
we have
4
x
4
y
2
≤
(
x
4
+
y
2
)
2
.
Conclude that
f
is continuous.
(b) For 0
≤
θ
≤
2
π
,
∞
< t <
∞
, define
g
θ
(
t
) =
f
(
t
cos
θ, t
sin
θ
)
.
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 Summer '11
 Staff
 Linear Algebra, Vector Space, Norm, Hilbert space, Banach space

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