Math 237
Assignment 2
Due: Friday, Sept 26th
1.
Determine if
f
is continuous at (0
,
0) where
f
(
x,y
) =
(
sin(
xy
)
ln(
x
2
+
y
2
+1)
if (
x,y
)
6
= (0
,
0)
0
if (
x,y
) = (0
,
0)
.
Solution:
f
(
x,y
) is not continuous at (0
,
0), since if we approach the limit along
y
=
x
we
get (using Taylor series)
lim
(
x,y
)
→
(0
,
0)
f
(
x,y
) = lim
x
→
0
sin
x
2
ln(2
x
2
+ 1)
= lim
x
→
0
x
2

x
6
6
+
···
2
x
2

(2
x
2
)
2
2
+
···
= lim
x
→
0
1

x
4
6
+
···
2

2
x
2
+
···
=
1
2
,
(L’Hospital’s rule would also work), which is not equal to
f
(0
,
0), thus
f
is not continuous
at (0
,
0).
2.
Determine where the function
f
(
x,y
) =
p
x
2

y
2
is continuous.
Justify your answer.
Solution: From assignment 1, we know that
f
(
x,y
) is not deﬁned at every point in any
neighborhood of points for which
x
2

y
2
= 0. Hence, by deﬁnition,
f
(
x,y
) can not be
continuous there.
For any point (
x,y
) satisfying
x
2

y
2
>
0, we have that
f
(
x,y
) is continuous by the
continuity theorems.
3.