Example 2.21.  Show that the function    is discontinuous along the negative real axis.  

Solution.  Let    denote a negative real number.  We compute the limit as z approaches through the upper half-plane and the limit as z approaches through the lower half-plane .  In polar coordinates these limits are given by    

            ,  and  

            .  

As the two limits are distinct, the function is discontinuous at .

Remark 2.4  Likewise,   is discontinuous at .  The mappings  ,  ,  and the branch cut are illustrated in Figure 2.18.

Explore Solution 2.21.

Enter the function    and find the limits as z approaches a point on the negative x-axis.

Since the two limits are different,    is discontinuous at all points    along the negative real axis.

                (a)  The branch      (where   ).   

                (b)  The branch     (where   ).  

            Figure 2.18  The branches    and    of  .   

(c) 2006 John H. Mathews, Russell W. Howell