# Measuring Fractal Dimension ?

Jaime Fernandez del Rio jaime.frio at gmail.com
Wed Jun 17 08:26:27 EDT 2009

```On Wed, Jun 17, 2009 at 1:52 PM, Mark Dickinson<dickinsm at gmail.com> wrote:
> Maybe James is thinking of the standard theorem
> that says that if a sequence of continuous functions
> on an interval converges uniformly then its limit
> is continuous?

Jaime was simply plain wrong... The example that always comes to mind
when figuring out uniform convergence (or lack of it), is the step
function , i.e. f(x)= 0 if x in [0,1), x(x)=1 if x >= 1, being
approximated by the sequence f_n(x) = x**n if x in [0,1), f_n(x) = 1
if x>=1, where uniform convergence is broken mostly due to the
limiting function not being continuous.

I simply was too quick with my extrapolations, and have realized I
have a looooot of work to do for my "real and functional analysis"
exam coming in three weeks...

Jaime

P.S. The snowflake curve, on the other hand, is uniformly continuous, right?

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```