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upgrayed 發問時間: 科學數學 · 9 年前

continuous functions

determine whether the function f(x) = 1/x , x ∈ R-{0} is continuous.

已更新項目:

the domain of f is the set R - {0} , the union of the two open intervals (-inf,0) and (0,inf) .

2 個已更新項目:

hint:

let {x_n} be any sequence in R - {0} that converge to c

2 個解答

評分
  • 9 年前
    最佳解答

    f is not continuous at 0 證明如下:

    取一點列{x_n}=1/n 1/n-->0 as n->∞

    但f(x_n)=n-->∞ as n->∞

    so f is not continuous at 0

    很明顯的 f is continuous on R\{0}

    2010-11-22 19:46:29 補充:

    因為f(0) is not defined

    2010-11-22 19:50:45 補充:

    其實這應該不用證明 很明顯

    2010-11-22 23:37:02 補充:

    given any c in R\{0}

    If x_n->c as n->∞

    then we have 1/x_n ->1/c

    =>f(x_n)->f(c)

    =>f is continuous on R\{0}

    參考資料: 以前所學
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  • 9 年前

    c is an arbitrary point of R-{0} , 這邊是否解釋一下會比較完善? 謝謝囉

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