Measuring Fractal Dimension ?

[Charles Yeomans]

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?

Charles Yeomans