Measuring Fractal Dimension ?
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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