※ 引述《GHJK777 (GHJK777)》之銘言：
: 標題: [微積] 雙變數證明一題(贈200P)
: 時間: Wed May 18 23:12:40 2011
:
: Prove that if f is a function of two variables that is
:
: differentiable at (a,b) then f is continuous at (a,b)
Since f is differentiable at (a,b),
there exists a unique vector grad(f) such that
f(a+h_1,b+h_2) - f(a,b) = grad(f(a,b))‧h + o(h),
where h=(h_1,h_2).
|f(a+h_1,b+h_2)-f(a,b)|
= |grad(f(a,b))‧h + o(h)|
<= |grad(f(a,b))‧h)| + |o(h)|
|grad(f(a,b))‧h| <= ∥grad(f(a,b))∥∥h∥
RHS→0 as h→0
and since lim(h→0)o(h)=0,
|im(h→0)|f(a+h_1,b+h_2)-f(a,b)|=0
=>lim(h→0)f(a+h_1,b+h_2)-f(a,b)=0
=>lim(h→0)f(a+h_1,b+h_2)=f(a,b)
Therefore f is continuous at (a,b).
:
: 希望可以用大一微積分的角度來解釋這題
:
: 謝謝
:
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: ◆ From: 140.112.24.201
: → lukqwertyuio:不連續就不可微分不是嗎?可微分所以連續。 05/18 23:27
: → mk426375 :課本沒有? 05/18 23:32
: → GHJK777 :一樓說的沒錯 但是怎麼證明呢？ 05/18 23:41
: → GHJK777 :課本似乎沒有 05/18 23:42
: 推 j0958322080 :哪本?? 05/18 23:49
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◆ From: 140.114.201.140
※ 編輯: mk426375 來自: 140.114.201.140 (05/19 00:08)
※ 編輯: mk426375 來自: 140.114.201.140 (05/19 00:11)