[微積] 為何sin(1/x)是integrable?

[Qmmmmnn (Qmmmmmmmmm)]

因為用手機發文，所以排版不太好，還請各位多多包涵...

最近讀到
lebesgue integrability and Riemann integrable
有個範例是：

The function f:(0,1) -> R defined by

f(x) = sin(1/x)

Is bounded and continuous, and therefore integrable,
On (0.1). But it is not piecewise
continuous because f(0+) does not
exist.


我覺得在x->0+的跳動很大，不知會是靠近1還是-1，所以總覺得沒辦法積分，不知道這跟下列這段有沒有關，因為我看不太懂

the simple criterion for integrability given by Lebesgue:
A subset of R is said to have measure zero if and only if it can be enclosed  in a finite or infinite sequence of open intervals whose combined total length - the sum of a finite or infinite series whose terms are the lengths of the individual intervals - is arbitrarily small, that is, smaller than any press signed positive number. Then Legesgue showed that f is Riemann integrable on (a,b) if and only if the set of points where f is discontinuous has measure zero.

麻煩各位了~謝謝~




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推 znmkhxrw    :一維的黎曼積分是定義在閉區間,勒貝格積分可以定義
→ znmkhxrw    :在任何可測集與可測函數
→ znmkhxrw    :你這個sin(1/x)自己在0那點定義一個值 則不連續點只
→ znmkhxrw    :有{0},測度0, 且sin(1/x)在[0,1]有界 所以黎曼可積
→ yhliu       :引用 Lebesgue 的定理當然是最直接. 不過, 由積分的
→ yhliu       :定義加上 sin(1/x) 在任何不含 0 的閉區間 [a,b],
→ yhliu       :0<a<b, 連續, 也是很容易的, 而且也能說明為何在靠近
→ yhliu       :0 時 sin(1/x) 的無限振盪不影響其可積性.
→ Qmmmmnn     :謝謝樓上各位的回答....