Exercise 17.  Let  ,  where  .  Show that    is discontinuous at each point along the negative x-axis.  

Solution 17.

See text and/or instructor's solution manual.

Solution.  Rewrite    as in Exercise 11, and mimic the solution with an arbitrary negative real number    taking the role of  .

Approach    from the upper half-plane.  If    along the upper semicircle  ,  then

          

Now approach     from the lower half-plane.  If    along the upper semicircle  ,  then

          

We are done.   

Aside.  We can let Mathematica double check our work.

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell