Example 3.6.  Show that the function defined by

              

is not differentiable at the point even though the Cauchy-Riemann equations are satisfied at .

Explore Solution 3.6.

Thus the Cauchy-Riemann equations hold at the origin.  

The two limits are distinct, so f is not differentiable at the origin.

We are done!

Aside.  We can also explore another solution.

Enter the function f[z].

Find the real and imaginary parts of  f[z].

Determine where the Cauchy-Riemann equations hold.

These equations are simultaneously zero, only when both    which appears to be only when    or at the origin.   But since they are not defined at the point , we must use limits to complete our investigation.

Let's check to see if the Cauchy-Riemann equations hold at .

We have shown that , that is, the Cauchy-Riemann equations do hold at the origin.

But,  f(z)  is not differentiable at because the following two limits are distinct.

The two limits are distinct, so we have shown that    is not differentiable at the point even though the Cauchy-Riemann equations are satisfied at .

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