Functions Without a Second DerivativeDate: 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/ |
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