REMOVABLE AND NONREMOVABLE DISCONTINUITY

                                                                                    

CONTINUITY

IF Lim f(x)= Lim  f(x) =  f(a)

    x->a-         x->a+

THEN

  

FUNCTION IS CONTINUOUS AT 'a'

WHERE  ,    Lim  f(x) is a left hand limit

                      x->a-

                      Lim f(x) is a right hand limit

                      x->a+

                          REMOVABLE DISCONTINUITY

IF Left hand limit = Right hand limit

But   f(a) is different; then it is called as Removable Discontinuity

So;       Lim f(x)= Lim f(x) but not equal to f(a)

            x->a -       x->a+

Then f(x) has removable discontinuity

WE CAN REMOVE  THIS DISCONTINUITY BY DEFINING 

VALUE OF F(a)

EX:

                                           

1.    Let   f(x)= 2x.x      ,   for  1≤ x<3

                     =  20       ,   for x=3

                     =6x         ,   for 3<x≤5

This is example of removable discontinuity

because;                         

                 Lim f(x)=2x.x=2.3.3=18

                 x->3-

                 Lim f(x)=6x=6.3=18

                 x->3+

 But                f(3)=20

                  so;   left hand limit and right hand limits are same(18)

                          but f(3) is different (20).

                  Hence function is not continuous at 3.

 IF WE DEFINE F(3)=18; LEFT HAND LIMIT , RIGHT HAND

LIMIT AND F(3) ARE SAME.AND FUNCTION BECOMES 

CONTINUOUS AT 3.SO THIS IS CALLED REMOVABLE DIS-

CONTINUITY.

                     NON-REMOVABLE DISCONTINUITY

If left hand limit , right hand limit at point say, 'a' and f(a)

are different; then this is non- removable discontinuity.

we cannot make this function continuous by redefining f(a)

EX:

F(X)=2   , X<10

        =3   , X=10

        =5   , X>10

 IN THIS EXAMPLE;

Lim F(x)=2

x->10-

Lim F(x)=5

x->10+

and F(10)=3.

 

as          2≠3≠5

then  function is not continuous

this is called as non removable discontinuity.

we cannot fix a single value of F(10) as there are different values of

left hand limit and right hand limit.(2 and 5).