中原大學九十一學年度博士班招生考試

91年 6月19日　08:30~10:00　　應用數學系 (選考)　　　 誠實是我們珍視的美德，
我們喜愛「拒絕作弊，堅守正直」的你！
科目： 分　　析  


Part A. 　Advanced Calculus
   
I.

Assume that and that is a convex function;-i.e., satisfies :

whenever and .
   
(a) (9%) Prove that, if , then .
   
(b) (8%) Prove that, for each , both and exist.
   
(c) (9%) Let A = { : is not differentiable at x}. Prove that A is at most countable.
   
   
II.
 (24%) Determine the truth or falsity of each of the following statements. (Give your reasons.)
   
(a) If : (0,1) →R　is differentiable on (0,1), then ' is continuous on (0,1).
   
(b)

If : R→R　is continuous and satisfies:

wherever and ,

then is uniformly continuous on R.
   
(c) If : R 2→R　has its partial derivatives (0,0) and (0,0) at the origin, then is continuous at the origin.
   
   
Part B. 　Real Analysis
   
I.
(20%) Assume that L P (R), where . For h R , define h(x) = (x+h). Prove that
   
  (a)
   
  (b)
   
   
II.
(15%) Suppose (X,,) is a measure space and n ,L1 (). Prove that, if , then {n} contains a subsequence {nk} which converges to almost everywhere.
 
   
   
III.
(15%) Let m denote the Lebesgue measure on R 1. Suppose that E is a Lebesgue measurable subset of R 1 and that m(E) > 0. Prove that there exist a > 0 such that .
 
   

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