Example 3.5.  Show that      is nowhere differentiable.

Solution.  We have   ,   where

and     .

Thus, for any point  ,

and     .

The Cauchy-Riemann equations (3-16) are not satisfied at any point  ,  so we conclude that

is nowhere differentiable.

Explore Solution 3.5.

Solution.  We have   ,   where

and     .

Thus, for any point  ,

and     .

The Cauchy-Riemann equations (3-16) are not satisfied at any point   ,   so we conclude that

is nowhere differentiable.

We are done.

Aside.  Both and can assist us in finding the partial derivatives.

Aside.  The Mathematica solution uses the commands.

``````

```

```

``````

``````

```

```

```

```

Aside.  The Maple commands are similar.

>

>

>

>

Thus,

.

The Cauchy-Riemann equations (3-16) are not satisfied at any point   ,   so we conclude that

is nowhere differentiable.

Remark.  In this book the use of computers is optional.

Hopefully this text will promote their use and understanding.

(c) 2011 John H. Mathews, Russell W. Howell