[分析] 幾題高微

[cherry770519 (＊桃子＊)]

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  1. If f: R   -> R is of class C   show that f is not 1-1


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  2. If f: R -> R   is of class C   show that f does not carry R onto R

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     in fact, show that f(R) contains no open subset of R


  3. Given a sequence (Xn) in R,define

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    lim Xn := lim sup{Xk , k >= n}  lim Xn := lim inf{Xk , k >= n}
              n->無限大             ---       n->無限大


    (a) find  the lim Xn and lim Xn of the sequence Xn= sin(n*(pi)/4)

    (b) For any bounded sequence (Xn) , show that

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        x= lim Xn  if and only if for any @ > 0 ,


       there is only finitely many Xn's satisfying Xn > = x + @

         but there is infinitely many Xn's satisfying Xn > = x - @








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