遊伊 發問時間: 科學數學 · 1 0 年前

請問一題高等微分證明

原題:Show that if fis bounede and monotone on( a , b ), then

lim(x↓a)f(x) and lim(x↑b)f(x)

Both exist

有證明提示:

shot that if f is monotone increasing, lim(x↓a)f(x)=inf(a,b)f

懇謝解答

2 個解答

評分
  • 1 0 年前
    最佳解答

    (proof):

    因為f在(a,b)上為單調有界函數,不妨假設f為單調遞增函數,單調遞減的證明亦同

    我只證明lim(x->a+)f(x)存在,lim(x->b-)的證明相似

    令S={f(x):x€(a,c]},where a<c<b

    Since f is monotone increasing function,f(a)<=f(x),for all x€(a,c]

    S is not empty,since c€S,S has a lower bounded f(a)

    so S has a greast lower bound,say α

    that is , α=inf{f(x):x€(a,c]}

    Now geven any ε>0,choose y€(a,c] such that f(y)<α+ε

    set δ=y-a,then for a<x<a+δ

    we have f(a)<=f(x)<=f(a+δ)=f(y)<α+ε

    so lim(x->a+)f(x)=α(右極限的定義)

    參考資料: me
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  • 1 0 年前

    很詳細!!

    我會努力研究的

    謝謝你喔~~

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