Example 3.9.  Show that the function    is differentiable at points that lie on the x and y axes but analytic nowhere.

Explore Solution 3.9.

Enter the functions u[x,y]  and  v[x,y] and use them to form  f[z].  Then determine where the Cauchy-Riemann equations hold.

The Cauchy-Riemann equations hold only if  x y = 0  which occurs along the coordinate axes.  Furthermore, all of the partial derivatives  , , , and   are continuous.  Hence,   is differentiable only when  ,  which occurs at points that lie on the coordinate axes.  Therefore,    is nowhere analytic.

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