Exercise 11.  Show that the function    is differentiable only at the point  .  

Hint.  To show that f is not differentiable at  ,  choose horizontal and vertical lines through the point and show that    approaches two distinct values as    along those two lines.  

Solution 11.

See text and/or instructor's solution manual.

Solution.  For the point    we compute  

                      

which shows that  .

Follow the hint for the rest.

Let    approach    along lines parallel to the x-axis and y-axis, respectively, giving:  

              

                         and  

             

At the point  ,  we have .

Hence, the limits are not equal, so    is not differentiable at the point  .  

This solution is complements of the authors.

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