推 k6416337:感謝!沒想到這個 早知道當初就讀Apostol了 要考試了 03/04 19:11
※ 引述《k6416337 (第一次獻給了涼宮版-光希)》之銘言：
: 1.Consider the sequence of functions
: fn(x)=╴╴x╴╴ for all x in |R,n=1,2,3,...
: 1+nx^2
: (i)Show that {fn} converges uniformly on |R to a differentiable function f.
: (ii)Show that f'(x)≠lim fn'(x) at some point in |R.
: n→∞
: 第(i)題我不知道該怎麼證，要怎樣變化才可以讓它只跟n有關?
: (ii)的話我會，f'(0)=0≠lim fn'(0)=1.
Proof. Notice that for any fixed x we have lim f_n(x)＝0. Let us consider
{1 ＋ n(x^2)} ≧ 2(√n)(|x|) by A.P. ≧ G.P.
Therefore, we have |f_n(x) － 0| ≦ 2/√n for all x in |R.
That is, f_n → f uniformly on |R.
1 － nx^2
On the other hand, f'≡ 0 on |R by f ≡ 0 on |R. And f'_n(x) ＝ ----------- .
(1＋nx^2)^2
So, we have f'_n(0) ＝ 1, which implies lim f'_n(0) ＝ 1 ≠ 0. In addition,
for any x ≠ 0, we have f'_n(x) → 0 by
1 － n x^2
------------------------ → 0.
n^2 x^4 ＋ 2n x^2 ＋ 1
Therefore, we have the conclusion: f'(x) ＝ lim f'_n(x) is correct if x ≠ 0,
but false if x ＝ 0.
NOTE.
(1) Rudin, 高微, Ch7. 第七題。
(2) 判斷均勻收斂與否，我們有許許多多的方法，但最好還是從定義去著手，然後考慮
其變形，兩本高微名著：Apostol 與 Rudin 習題最好每一題一定要作一次!
Sequence (Series) of functions
在分析學裡可是有極端重要的地位。
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