[試題] 103-2 劉豐哲 實分析二 期中考

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課程名稱︰實分析二
課程性質︰數學研究所必選修
課程教師︰劉豐哲
開課學院：理學院
開課系所︰數學系、數學研究所、應用數學科學研究所
考試日期（年月日）︰104.4.22
考試時限（分鐘）：100min

試題 :

1.(10%) Let f be a BV function on [a,b]. Show that f is AC if and only if

   b      b
  V (f)＝∫|f'|dλ.
   a      a

2.(15%) Let f be a continuous function defined on a finite open interval (a,b)

  such that f is AC on every closed interval in (a,b) and f' is integrable on

  (a,b). Show that f can be extended to be an AC function on [a,b].

                        n
3.(10%) Suppose K⊂Ω⊂R, where K is compact and Ω is open. Show that there

         ∞
  is u∈C (Ω) such that 0≦u≦1 and u＝1 on K.
         c

                                                                       n
4.Let f be a locally integrable function defined on an open set Ω in R.

                                              n
  (i)(10%) Show that f＝0 a.e. on Ω if ∫fdλ ＝0 for all compact set K in Ω.
                                 n       K         ∞

  (ii)(10%) show that if ∫ fφdλ ＝0 for all φ∈C (Ω), then f＝0 a.e. on Ω.
                          Ω                       c

5.Suppose that t is a  continuously differentiable map from an open set Ω in

   n      n
  R into R.

                       n
  (i)(10%) Show that λ (tA)＝0 if A is a null set in Ω.


  (ii)(12%) Let D={x∈Ω：J(t;x)=0} , where J(t;x) is the Jacobian determinant

                             n
      of t at x. Show that λ (tD)=0.


6.(8%) Let f be the Cantor's ternary function on [0,1] and P the Cantor's tern-

                          1
  ary set in [0,1]. Find ∫fdf and ∫fdμ , where μ is the Lebesgue-Stieltjes
                          0         P   f          f

  measure on [0,1] generated by f.


7.(15%) Show that

                  t 　2x        ∞    j t  2j+1
                 ∫  一一  dx＝2Σ(-1) ∫ x    dx
                  0  1+x^2     j＝0     0

  for 0＜t＜1 ; then show that

                                1  2x      ∞    j 1
                               ∫ 一一 dx＝Σ(-1) 一一
                                0 1+x^2   j＝0    j+1

               ∞    j 1
  and evaluate Σ(-1) 一一.
              j＝0    j+1

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※ 編輯: dddlabc (140.112.244.39), 04/23/2015 17:15:40
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