[分析] 關於課本Wade裡的三個定理

[iamagine (A-gine)]

課本：[Wade]

Theroem11.13
Let f be a vector function.
If f is differentiable at a ,then f is continuous at a.

Theroem11.14
Let f be a vector function.
If f is differentiable at a
then all first-order partial derivatives of f exist at a.

這裡為什麼只能得到first-order partial derivatives存在?
而不是得到first-order partial derivatives存在而且在a點連續呢?


Theorem11.15
Let V be open set in |R^n
Let a E V   (a屬於V)
Let f:V→|R^m
If all first-order partial derivatives of f exist in V and are continuous at
a,
Then f is differentiable at a.

這裡的條件是要first-order partial derivatives在a點
不僅僅要存在，而且first-order partial derivatives在a點還要連續.

我想問的是，為什麼定理15的條件要強到：它偏導數在a點要連續?
只是存在的話，f在a點就不可微了嗎?
(如果可以，能不能麻煩大大舉實際的例子跟我說>""<)

麻煩大大了~

謝謝大家~

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From:一直都學不好的我

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