3.2 單邊連續

 

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 3-2  單邊連續

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在閉區間有定義,如果在開區間是連續且

  

在閉區間上是連續

此外,點是右連續

                點是左連續 

 

1.      請問是否連續?   

 

解答:

(1)      存在

(2)  

            

由上式可得:      

(3) 但是   可知

 

 

2.      ,若要求處連續,則值應為何?

 

 

 

 

解答:

連續定義             

左極限                 

   右極限                  

 

3.    ,且對所有實數皆為連續, 求c ,d =?

 

 

解答:

處,需滿足  

即得條件一      

                

兩者相等得      

處,需滿足   

即得條件二      

                

兩者相等得      

聯立解得        

 

 

 

 

4.      Show that is continuous on.  

  

解答:

  By definition of continuity

 

   is continuous on

 

 

    is continuous on

 

  Thus,  exist and equals 1.

  Also,

  Thus, is continuous at .

  We concluded is continuous on.

 

 

 

  

5.      Show that is continuous on

  

 

解答:

  By definition of continuity

 

    is continuous on

 

 

    is continuous on

 

  Thus,  exist and equals .

  Also,

  Thus, is continuous at .

  We concluded is continuous on.

 

 

  

6.      Find the numbers at which  is discontinuous. 

  

 

 

解答:

  is continuous on , and  since it is a polynomial on each of these intervals.

 

    is continuous on

 

 

   is continuous on

 

  Thus,  doesn’t exist at 0.

  Since ,  is continuous from left at 0

   is discontinuous at 0.

 

  Also, , , and

  Thus, is continuous at .

  The only number at which  is discontinuous is .

 

 

   

7.      Find the numbers at which  is discontinuous. 

  

解答:

  is continuous on , and  since it is a polynomial on each of these intervals.

  , , so  is discontinuous at .

  , , so  is discontinuous at .

  Since ,  is continuous from left at 1.

                                    

 

 

8.      Find the numbers at which  is discontinuous. 

  

   解答:

  is continuous on , and  since it is a polynomial on each of these intervals.

  , , so  is discontinuous at .

  ,  is continuous from left at 1.

  , , so  is discontinuous at .

  Since ,  is continuous from the right at 3.

                                                            

 

 

  

9.      For what value of constant c is the function continuous on

  

解答:

  is continuous on and .

  By definition of continuity

 

   is continuous on

 

 

    is continuous on

 

  So  is continuous at .

 

 

  Thus, for  to be continuous on, .

 

 

 

 

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