Measuring Fractal Dimension ?

Paul Rubin http
Wed Jun 17 08:04:11 CEST 2009


Jaime Fernandez del Rio <jaime.frio at gmail.com> writes:
> I am pretty sure that a continuous sequence of
> curves that converges to a continuous curve, will do so uniformly.

I think a typical example of a curve that's continuous but not
uniformly continuous is

   f(t) = sin(1/t), defined when t > 0

It is continuous at every t>0 but wiggles violently as you get closer
to t=0.  You wouldn't be able to approximate it by sampling a finite
number of points.  A sequence like 

   g_n(t) = sin((1+1/n)/ t)    for n=1,2,...

obviously converges to f, but not uniformly.  On a closed interval,
any continuous function is uniformly continuous.



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