[微積] continuous derivative 問題

[anoymouse (沒有暱稱)]

Thomas' calculus 關於積分弧長，有先做一個前提:
Suppose the curve whose length we want to find is the graph of the function
y = f(x) from x= a to x = b.
In order to derive an integral formula for the length of the curve, we assume
that ƒ has a continuous derivative at every point of [a, b]. Such a function
is called smooth,and its graph is a smooth curve because it does not have any breaks, corners,
or cusps.

最後說到:
(1 + [f'(x)]^2)^1/2 is continuous on [a,b]


想問f has a continuous derivative ... of [a,b]的意思是
(1) f'(x) is continuous on [a,b] 還是
(2) f在[a,b]每一點都有derivative

如果是(2) 舉一個example https://imgur.com/Wv2wDLl
g(x)每一點都有derivative 但是 g'(0)在x = 0 並不連續。
這樣的話 "(1 + [f'(x)]^2)^1/2 is continuous " 就不一定會成立吧?

所以原文的意思是應該是(1)?

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推 cuteSquirrel: 您po的原文敘述有強調是平滑曲線，排除斷點 折角
→ cuteSquirrel: "not have any breaks, corners or cusps."
→ anoymouse   : 所以你認為是（2）？
→ anoymouse   : 我的舉例g of x雖然靠近0會劇烈跳動 但應該每個點都
→ anoymouse   : 可以微分且g'在0不連續吧 這樣不符合圓滑曲線嗎
推 cuylerLin   : 這是英文問題不是數學問題，a continuous derivativ
→ cuylerLin   : e 的意思是 a derivative that is continuous 也就
→ cuylerLin   : 是導數是一個連續函數。
推 arrenwu     : 就 (1)
→ cmrafsts    : 我還以為是要問在端點微分是什麼意思XD
→ anoymouse   : 好的 謝謝各位
推 alan23273850: 當然是 1，語意蠻明確的