(ii)  Consider the function   if is rational, and if is irrational. It is discontinuous at every       point. It has essential discontinuity at every point since limit at every point does not exist.

(iii) The function
    
     is discontinuous at .Once again it is a point of essential discontinuity.
      

Practice Exercises 6.1: Discontinuities of a function

1. Analyse the points of discontinuity and the types of discontinuity for the following functions:
                      
      

2. Find the type of discontinuity has at :
(i) .
(ii) .

3. Let
    Show that is discontinuous at every  Analyse the type of discontinuity also.