Measuring Fractal Dimension ?
charles at declareSub.com
Thu Jun 18 14:32:13 EDT 2009
On Jun 18, 2009, at 2:19 PM, David C. Ullrich wrote:
> On Wed, 17 Jun 2009 07:37:32 -0400, Charles Yeomans
> <charles at declareSub.com> wrote:
>> On Jun 17, 2009, at 2:04 AM, Paul Rubin wrote:
>>> 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
>>> 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.
>> Isn't (-?, ?) closed?
> What is your version of the definition of "closed"?
My version of a closed interval is one that contains its limit points.
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