Measuring Fractal Dimension ?

[Charles Yeomans]

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  
>>> 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.
>>
>> 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.

Charles Yeomans