Exercise 7.  Use any method to show that the following functions are nowhere differentiable.  

7 (b).  .

Solution 7 (b).

See text and/or instructor's solution manual.

Answer.  ,   ,   ,   .

The Cauchy-Riemann equations hold if and only if both  ,  which is impossible.

Therefore,     is nowhere differentiable.  

Solution.  ,   and

                 ,           so that  

,     ,     ,     .   

The  Cauchy Riemann equations are  

        ,  

        ,

the equation    is impossible.

Therefore,     is nowhere differentiable.  

We are done.   

Aside.  A reason why h(z) is not analytic is given in Exercise 16.  

     Loosely speaking, if h(z) is analytic then there cannot be any occurrence of the variable  .

So technically there are "hidden" terms involving    in the formula for  g(z),  and this precludes the possibility that  g(z)  can be analytic.

Indeed, the complex form of the Cauchy-Riemann equations fails to hold because  .  

The function    is real valued and the image of any region would be a segment on the real axis and is not an interesting graph.  

This solution is complements of the authors.

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