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SOLUTIONS FOR ASSIGNMENT 3
Problem Set 2, A6: Let
f
(
x, y
) =
x

y
−
1

. Use the definition to prove that
f
is not differentiable at
(1
,
1)
.
Note that the point
(1
,
1)
is a typical point as regards differentiation w.r.t
x
but
is an exceptional point as regards differentiation w.r.t.
y
.
Since
∂f
∂x
=

y
−
1

, it follows that
∂f
∂x
(1
,
1) = 0
.
Consider now
lim
h
→
0
f
(1
,
1 +
h
)
−
f
(1
,
1)
h
= lim
h
→
0

h

h
=
+1
,
if
h >
0
=
−
1
,
if
h <
0
.
The above limit does not exist and thus
∂f
∂y
(1
,
1)
does not exist either. This im
plies that
f
is not differentiable at
(1
,
1)
, by the definition of differentiability.
Problem Set 2, B7(i): Conside the function
f
(
x, y
) = (
xy
)
2
/
3
.
a). Use the definition to determine whether
f
is differentiable at
(0
,
0)
.
b). Using the answer to part a), can you use one of the theoretical results to
draw a conclusion concerning the continuity of
f
at
(0
,
0)
?
c). Using the answer to part a), can you use one of the theoretical results to
draw a conclusion concerning the continuity of
f
x
and
f
y
at
(0
,
0)
.
a). Note that
f
is symmetric in
x
and
y
and that
(0
,
0)
is an exceptional point
as regards differentiation w.r.t both
x
and
y
. We need to use the definition of dif
ferentiability to check whether
f
is differentiable at
(0
,
0)
.
Consider
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 Spring '08
 Oancea
 Math

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