Functions Without a Second Derivative

Date: 6/28/96 at 11:9:0
From: Anonymous
Subject: Functions Without a Second Derivative

What are some examples of functions of a real variable whose 
derivatives don't have derivatives? This seems to be the main 
difference between real functions and complex functions which seem to 
have derivatives of every degree.

Date: 6/28/96 at 18:10:52
From: Doctor Tom
Subject: Re: Functions Without a Second Derivative

Let f(x) = 0 if x <= 0 and let f(x) = x^2, for x > 0.  
At the point x=0, the derivative exists (it's zero), and the second 
derivative does not exist, since it is the function:
f'(x) = 0, for x <= 0 and f'(x) = 2x, for x > 0.  

The derivative has a sharp corner at x=0, so there's no second 
derivative.

As you've noticed, this is a major difference between differentiable
functions in the real and complex plane.  The condition of being
differentiable is MUCH stronger in the complex case, and one
can show that if a function has one derivative, it has derivatives
of all orders.

Maybe the way to see why it might be true is to look at the
definition of a derivative.  We say a function has a derivative if:

    lim (h->0)  (f(x+h)-f(x))/h exists.

In the case of the real numbers, h can only approach zero from
two directions - the positive and negative sides.  In the
case of complex functions, this limit has to hold, no matter
how h goes to zero - along any direction, spiraling in, or
whatever.  This is a far stronger condition on f(x).

-Doctor Tom,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/