The Real And Complex Number Systems

So, we have
Partial derivatives
fb fa b af x.
5.33 Consider the function f defined on R 2 by the following formulas:
xy
fx, y if x, y 0, 0 f0, 0 0.
x 2 y2 Prove that the partial derivatives D 1 fx, y and D 2 fx, y exist for every x, y in R 2 and
evaluate these derivatives explicitly in terms of x and y. Also, show that f is not continuous
at 0, 0.
Proof: It is clear that for all x, y 0, 0, wehave
D 1 fx, y y y2 x 2
x 2 y 2 and D 2fx, y x x2 y 2
2 x 2 y 2 . 2
For x, y 0, 0, wehave
fx,0 f0, 0
D 1 f0, 0 lim
0.
x0 x 0
Similarly, we have
D 2 f0, 0 0.
In addition, let y x and y 2x, wehave
limfx, x 1/2 limfx,2x 2/5.
x0 x0
Hence, f is not continuous at 0, 0.
Remark: The existence of all partial derivatives does not make sure the continuity of f.
The trouble with partial derivatives is that they treat a function of several variables as a
function of one variable at a time.
5.34 Let f be defined on R 2 as follows:
fx, y y x2 y 2
if x,,y 0, 0, f0, 0 0.
x 2 y2 Compute the first- and second-order partial derivatives of f at the origin, when they exist.
Proof: For x, y 0, 0, it is clear that we have
4xy
D 1 fx, y
3
x 2 y 2 and D 2fx, y x4 4x 2 y 2 y 4
2 x 2 y 2 2
and for x, y 0, 0, wehave
fx,0 f0, 0
f0, y f0, 0
D 1 f0, 0 lim
0, D
x0 x 0
2 f0, 0 lim
1.
y0 y 0
Hence,
and
D 1,2 f0, 0 lim
x0
D 1 fx,0 D 1 f0, 0
0,
x 0
D 2 fx,0 D 2 f0, 0
lim 2
x 0
x0 x does not exist,
D 1 f0, y D 1 f0, 0
0,
y 0
D 1,1 f0, 0 lim
x0
D 2,1 f0, 0 lim
y0