3.1 連續定義

 

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3.1 連續定義
3.2 單邊連續
3.3  連續基本定理

 

 

3-1   連續定義

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連續之定義:

『若 (1) 有確切定義,

  (2) 也存在 ,

     (3)

則稱為連續的』。

 

 

連續函數五大運算之基本定理   

為連續及為連續,則五大運算後仍為連續。

 

 

 

 

 

1.  是連續函數,不是連續函數,,則

(A)     必都不連續

(B)       必不連續,有可能連續

(C)       *有可能連續,必不連續

(D)     *都有可能連續

 

解答:  (B)必不連續,有可能連續

                是連續函數,不是連續函數

                不是連續函數

                  是連續函數

 

 

 

2.      若函數  。 請問函數是否連續?

 

解答:

  (1)

  (2)       

  (3)

 根據連續基本定義:

3.      Use the definition of continuity to show that the function is continuous at the number a. .

 

解答:

  By definition of continuity

 

         

         

         

  So  is continuous at the number .

 

 

 

  

4.      Use the definition of continuity to show that the function is continuous at the number a. .

 

解答:

  By definition of continuity

 

     

     

     

     

  So  is continuous at the number .

 

 

 

 

5.      Use the definition of continuity to show that the function is continuous at the number a. .

 

解答:

  By definition of continuity

 

        

         

         

  So  is continuous at the number.

 

 

 

   

6.      Use the definition of continuity to show that the function is continuous at the interval. .

 

解答:

  By definition of continuity

  For

 

     

      

  So  is continuous at, for every  in .

  That is,  is continuous on .

 

 

 

 

7.      Use the definition of continuity to show that the function is continuous at the interval. .

 

解答:

  By definition of continuity

  For

 

     

      

  So  is continuous at, for every  in.

  That is,  is continuous on.

 

 

   

8.      Explain why the function is discontinuity at the given number a.         .

 

解答:

   is discontinuity at  since  is not define at

                                   

 

 

   

9.      Explain why the function is discontinuity at the given number a.         .

 

 

解答:

   is discontinuity at  since  does not exist.

                                       

 

 

 

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