DISCONTNUITY

 Defintition of Continuity :

     The function of the graph which is not connected with each other is known as a discontinuous function. A function f(x) is said to have a discontinuity of the first kind at x = a, if the left-hand limit of f(x) and right-hand limit of f(x) both exist but are not equal. f(x) is said to have a discontinuity of the first kind from the left at x = a, if the left hand of the function exists but not equal to f(a).

Discontinuity

In the above graph, the limits of the function to the left and to the right are not equal and hence the limit at x = 3 does not exist anymore. Such function is said to be a discontinuity of a function.

Types of Discontinuity :

There are three types of discontinuity such as :

  1. Jump Discontinuity 
  2. Removable Discontinuity .
  3. Infinite Discontinuity.
1.  Jump Discontinuity :
              A function f(x) has a jump discontinuity at x = a if the left hand limit and right hand limits both exist but are different , that is  lim x →a+ f(x) ≠ lim x →a- f(x) .
In this diagram , both left -hand limit and right-hand limits may exist but they are different . So the graph "jumps" at x = a . The function is said to have a jump discontinuity.
 
eg., Consider    f(x) = x2 -x -5  , for -4  ≤ x < -2.
                                =x3 - 4x - 3 , for -2  ≤ x  ≤ 1 .
for x = 2  , f(-2) = (-2)3 -4(-2) -3 = -3.
lim x →2- f(x) = lim x →2 f(x2 -x -5) = 4 + 2 - 5 = 1 and 
lim x →2- f(x) ≠ lim x →2 f(x3 - 4x - 3) = -8 + 8 -3 = -3 
lim x →2+ f(x) ≠ lim x →2- f(x) .
Hence lim x →2- f(x) does not exist .
∴The function f(x) has a jump discontinuity.


2.  Removable Discontinuity :

      A function f(x) has a discontinuity at x = a , and lim x →a f(x) exist , but either f(a) is not defined or   lim x →a f(x) ≠  f(a) . In such case we define f(a) as   lim x →a+f(x) . Then with new definition , the function f(x) becomes continuous at x = a . Such a discontinuity is called a Removable Discontinuity .                                     
In thid figure the function has a limit . Howere , there is a hole or graph at x = a , f(x) is not defined at x= a . That can be represented by defining f(x) at x= a .


  3. Infinite Disconitinuity :
        
     In Infinite Discontinuity, either one or both Right Hand and Left Hand Limit do not exist or are Infinite. It is also known as Essential Discontinuity. Whenever the graph of a function f(x) has the line x = k, as a vertical asymptote, then f(x) becomes positively or negatively infinite as x→k+ or x→k–. Then, function f(x) is said to have an infinite discontinuity.
    
Obser this graph xx = 1 and y =f(x) =1/x is the function to be considered . It is easy to see that f(x) →∞ as x →0+ and f(x) →∞ as x →0- . f(x) is not defined . Of course , this function is discontinuous at x = a.
A function f(x) is said to be have a infinite discontinuity at x =a ,
 if   lim x →a- f(x) ≠ ± ∞ or lim x →a+ f(x) = ± ∞ 
∴ the function f(x) has an infinite discontinuity.

 

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