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Real Analysis
book

Real Analysis

by V. Karunakaran
May 2024
Intermediate to advanced content levelIntermediate to advanced
585 pages
15h 38m
English
Pearson India
Content preview from Real Analysis
“real: chapter_12” 2011/5/23 1:04 page 45 #45
Real-valued Functions of Two Real Variables 12-45
However as h + ik 0 through the line x = y (i.e. h = k), we
have
lim
h0
f (h) f (0)
h
= lim
h0
h
2
(1 + i)
2
h
2
(1 + i)
2
=
2i
2i
=−1.
Hence f
(0) does not exist and by Note 12.8.6, f cannot be
complex differentiable.
3. Consider
f (x, y) =
x
3
x
2
+y
2
if (x, y) = (0, 0)
0if(x, y) = (0, 0).
|f (x, y) f (0, 0)|=
x
3
x
2
+ y
2
≤|x|→0as(x, y) (0, 0).
Thus f is continuous at (0, 0). Further
Df (0, α) = lim
h0
f (he
iα
) f (0)
h
= lim
h0
h
3
cos
3
α
h
3
= cos
3
α
exist for all α ∈[0, 2π) and in particular f
x
(0, 0) = 1 and
f
y
(0, 0) = 0. However f (x, y) is not real differentiable ...
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Publisher Resources

ISBN: 9781299447561Publisher Website