Example 3.2.  Show that the function   is nowhere differentiable.

Explore Solution 3.2.

First, form the difference quotient using  .  

And this limit is easy to compute.

Second, form the difference quotient using  .  

And this limit is easy to compute.

Since the two limits are different for any point    does not exist for any  .  

Aside.  Mathematica will not differentiate  .  

The limit definition has shown that   is nowhere differentiable.

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