[微積] 為何sin(1/x)是integrable?

看板Math作者Qmmmmnn (Qmmmmmmmmm)時間13年前 (2012/12/04 19:01)推噓1(1推 0噓 8→)

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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~ -- ※ 發信站: 批踢踢實業坊(ptt.cc) ◆ From: 49.215.39.234

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