# closure

• Mar 26th 2010, 06:51 AM
vinnie100
closure
How do you show that the closure of [a,b] is not closed in C0[a,b]?
• Mar 26th 2010, 07:51 AM
mabruka
C0[a,b] is .. the space of continuous functions with compact support..?

If the answer is affirmative then i cant help you, [a,b] is not a subset of C0[a,b] .

If the answer is negative then i cant help you since i dont know what you mean with C0[a,b].

Need context
• Mar 26th 2010, 08:55 AM
vinnie100
Sorry I got it all wrong!

Its meant to say:

Show that a p times continuously differentiable function is not closed in a 0 times continuous differentiable function.
• Mar 26th 2010, 11:12 PM
southprkfan1
Quote:

Originally Posted by vinnie100
Sorry I got it all wrong!

Its meant to say:

Show that a p times continuously differentiable function is not closed in a 0 times continuous differentiable function.

Assuming I follow you, consider the set of functions on [0,1]:

fn(x) = $x^{(p+n)} \forall n=1,2,3...$

These functions are clearly p times differentiable. But, as n goes to infinity, the functions converge to the function f(x) where

f(x) = 0 if $x\not=$ 1 and f(x) = 1 if x=1, which is not continuous
• Mar 26th 2010, 11:30 PM
Drexel28
Quote:

Originally Posted by southprkfan1
Assuming I follow you, consider the set of functions on [0,1]:

fn(x) = $x^{(p+n)} \forall n=1,2,3...$

These functions are clearly p times differentiable. But, as n goes to infinity, the functions converge to the function f(x) where

f(x) = 0 if $x\not=$ 1 and f(x) = 1 if x=1, which is not continuous

What did you think he meant? I don't actually understand what he was asking for.
• Mar 27th 2010, 01:01 AM
mabruka
Quote:

What did you think he meant? I don't actually understand what he was asking for.
I think what southprkfan1 thought vinnie100 meant was:

Prove that $\mathcal C^p([a,b])$ is not closed on $\mathcal C([a,b])$

where $C^p([a,b])$ is the set of functions with p continuous derivative on [a,b] and $\mathcal C([a,b])$ the set of continuous functions