[微積] continuous derivative 問題
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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