4. Let

    Show that is continuous only at  Analyse the type of discontinuities at other points.

5. Let have essential discontinuity at . What can you say about

Optional Exercise (Discontinuities of a monotone function):

Try to draw the graph of a monotone function with a discontinuity at a point .

You will realize that such function can have only jump discontinuities, and this is not difficult to prove. In fact, one can show that monotone functions can have at most ‘countable' number of discontinuities.