upgrayed 發問時間： 科學數學 · 10 年前

# continuous functions

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

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

• 10 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
• 10 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