作者mk426375 (时雨)
看板Math
标题Re: [微积] 双变数证明一题(赠200P)
时间Thu May 19 00:06:53 2011
※ 引述《GHJK777 (GHJK777)》之铭言：
: 标题: [微积] 双变数证明一题(赠200P)
: 时间: 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).
:
: 希望可以用大一微积分的角度来解释这题
:
: 谢谢
:
: --
:
※ 发信站: 批踢踢实业坊(ptt.cc)
: ◆ From: 140.112.24.201
: → lukqwertyuio:不连续就不可微分不是吗?可微分所以连续。 05/18 23:27
: → mk426375 :课本没有? 05/18 23:32
: → GHJK777 :一楼说的没错 但是怎麽证明呢？ 05/18 23:41
: → GHJK777 :课本似乎没有 05/18 23:42
: 推 j0958322080 :哪本?? 05/18 23:49
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※ 发信站: 批踢踢实业坊(ptt.cc)
◆ From: 140.114.201.140
※ 编辑: mk426375 来自: 140.114.201.140 (05/19 00:08)
※ 编辑: mk426375 来自: 140.114.201.140 (05/19 00:11)