[分析] discontinuities of a B.V. function

看板Math作者cyt147 (大叔)時間6年前 (2018/05/26 17:38)推噓3(3推 0噓 2→)

留言5則, 3人參與討論串1/1

大家好，有個問題想請教各位。我目前正試著證下列性質: Let f:[a,b]→R be of bounded variation. If f has a discontinuity, it must be a simple jump. Furthermore, the discontinuities of f form a countable set. NOTE: A simple jump has limits on both sides if it's an interior point, and has the one-sided limit if it's an endpoint. ============================================================================ 我已經知道: 1. A function of bounded variation can be expressed as the difference of two increasing functions. 2. An increasing functions can have a discontinuity only when it's a simple jump. Besides, the set of discontinuities is countable. 我查了幾本分析導論，都是用上面兩個lemma"說明"，沒有詳細過程。如果想寫個一清二楚，請問該怎麼下手?謝謝。我的開頭目前就只有: Suppose f=g-h for some increasing functions g,h. -- ※ 發信站: 批踢踢實業坊(ptt.cc), 來自: 180.177.114.46 ※ 文章網址: https://www.ptt.cc/bbs/Math/M.1527327499.A.601.html

推

LiamIssac

05/26 17:41, 6年前 , 1^F

05/26 17:41, 1^F

謝謝回應。後來想想，似乎沒什麼好證的。由極限的減法可知g-h在不連續點會有左、右極限，所以 f的不連續點必為simple jump。另一方面，在用g、h製作g-h的時候，不論他們的不連續點是否減少，終究各自成一個countable set，取聯集當然還是countable set。我覺得這個證明寫仔細的話大概就是這樣。 ※ 編輯: cyt147 (180.177.114.46), 05/26/2018 18:13:32 ※ 編輯: cyt147 (180.177.114.46), 05/26/2018 18:15:57

推

mike50378

05/26 18:42, 6年前 , 2^F

05/26 18:42, 2^F