How do you show that the closure of [a,b] is not closed in C0[a,b]?

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- Mar 26th 2010, 05:51 AMvinnie100closure
How do you show that the closure of [a,b] is not closed in C0[a,b]?

- Mar 26th 2010, 06:51 AMmabruka
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, 07:55 AMvinnie100
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, 10:12 PMsouthprkfan1
Assuming I follow you, consider the set of functions on [0,1]:

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

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 $\displaystyle x\not=$ 1 and f(x) = 1 if x=1, which is not continuous - Mar 26th 2010, 10:30 PMDrexel28
- Mar 27th 2010, 12:01 AMmabrukaQuote:

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

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

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