Continuous FunctionDate: 05/18/99 at 09:11:55 From: steve Subject: Abstract analysis I am interested to know if there is any way of constructing a continuous function that is nowhere differentiable. Could you give me an example of one such function? Date: 05/18/99 at 09:38:24 From: Doctor Mitteldorf Subject: Re: Abstract analysis Dear Steve, If you look in an introduction to Wavelet Theory, I think you'll find an abundance of such functions and their uses. Here's an example. On the domain 0 <= x <= 1, start with step 0, y = 0. Next step, create a wedge in the middle 1/2, lifting the midpoint by 1 unit 0) y = 0 1) y = 0 for x <= 1/4 y = 4(x-1/4) for 1/4 <= x <= 1/2 y = 4(3/4-x) for 1/2 <= x >= 3/4 y = 0 for x >= 3/4 For the next step, raise the midpoints of each of the four segments by 1/4 units, again dividing each of the 4 segments into 4. Continue this process ad infinitum, and you have a continuous function of x that is nowhere differentiable. - Doctor Mitteldorf, The Math Forum http://mathforum.org/dr.math/ |
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