推 GHJK777 :先搶先贏 P幣給囉！ 05/19 00:59
※ 引述《GHJK777 (GHJK777)》之銘言：
: Prove that if f is a function of two variables that is
: differentiable at (a,b) then 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
我用的是Salas的課本
定義如下
f:R^2--->R is diff. at x=(a,b) in R^2 if exists y in R^2 such that
f(x+h)-f(x)=y．h+o(h)
,where o(．) is little oh
Property of o(h)
o(h)--->0 as h--->0
And,in fact, y is grad(f)
|f(x+h)-f(x)|=|y．h+o(h)|≦|y．h|+|o(h)|≦||y||．||h||+|o(h)|--->0 as h--->0
therefore f(x+h)-f(x)--->0 => f(x+h)--->f(x)
therefore f is conti at x
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