[問題] 如何証明f(x,y)不可微?

[A-gine]

f(x,y)= { 0 if (x,y)＝(0,0)
|
| __(x^2)y__ if (x,y)≠(0,0)
{ (x^4)+(y^2)

証明：f(x,y)在(0,0)不可微.



它在(x,y)≠(0,0)時，那個分子和分母

沒辦法讓我按照可微的定義，然後令△y＝ m△x

結果消不掉…


麻煩大大們了～

謝謝大家


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[漫不經心]

Post by A-gine
f(x,y)= { 0 if (x,y)＝(0,0)
|
| __(x^2)y__ if (x,y)≠(0,0)
{ (x^4)+(y^2)
証明：f(x,y)在(0,0)不可微.
它在(x,y)≠(0,0)時，那個分子和分母
沒辦法讓我按照可微的定義，然後令△y＝ m△x
結果消不掉…
麻煩大大們了～
謝謝大家

We shall assume that f is differentiable at (0,0) and arrive at a

contradiction. Suppose f is differentiable at (0,0).

Then f is continuous at (0,0) by theorem. Now, consider

a sequences of points {a_n} = {(1/n, 1/n^2)} in |R^2,

it is clear that when {a_n} tend to (0,0) (as n -> ∞),

f({a_n}) -> 1/2 which is not 0 = f(0,0) and hence f is dicontinuous at (0,0).

This is a contradiction. From another point of view, if you approach (0,0)

along the parabola y = mx^2 where (x,y) is not (0,0), then

f(x,y) -> m/(1 + m^2) which depends on m as (x,y) tends to (0.0).

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※ Origin: 交大次世代(bs2.to)
◆ From: 123-195-218-100.dynamic.kbronet.c

[A-gine]

Post by 漫不經心

f(x,y)= { 0 if (x,y)��(0,0)
|
| __(x^2)y__ if (x,y)��(0,0)
{ (x^4)+(y^2)
�����Gf(x,y)�b(0,0)���i�L.

We shall assume that f is differentiable at (0,0) and arrive at a
contradiction. Suppose f is differentiable at (0,0).
Then f is continuous at (0,0) by theorem. Now, consider
a sequences of points {a_n} = {(1/n, 1/n^2)} in |R^2,

^^^^^^^^^^^^^^^^^^^^^^
�j�j�A�o�ӼƦC�������O?= =a

Post by 漫不經心
it is clear that when {a_n} tend to (0,0) (as n -> ��),
f({a_n}) -> 1/2 which is not 0 = f(0,0) and hence f is dicontinuous at (0,0).
This is a contradiction. From another point of view, if you approach (0,0)
along the parabola y = mx^2 where (x,y) is not (0,0), then

 ^^^^^^^^^^�j�j�A�o�ӭn�����O�O?�q�����ݥX�Ӫ�= =a
�o���Ӧa�褣����A�·Фj�j�F~

Post by 漫不經心
f(x,y) -> m/(1 + m^2) which depends on m as (x,y) tends to (0.0).

--

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[漫不經心]

二個方法的技巧都是讓原來題目中的 f(x,y) 的分子和分母

都變成四次的齊次式(即各項都是 4 次式)，這樣才能讓不連續的可能性增加。

至於第一個方法除了是強迫配成齊次式之外，還要選一個數列 {a_n}

其極限值是 0，且讓 f({a_n}) 的極限值不是 0，

而 {(1/n,1/n^2)} 就是一種選法。

--
※ Origin: 交大次世代(bs2.to)
◆ From: 123-195-218-100.dynamic.kbronet.c