Is C^l closed in C^0?

1. Apr 12, 2012

SP90

1. The problem statement, all variables and given/known data

Is $C^{k}[a,b]$ closed in $C^{0}[a,b]?$

3. The attempt at a solution

$C^{k}[a,b]$ is obviously a subset of $C^{0}[a,b]$.

My gut feeling says no. I thought the best way would be to construct a function $f_{n}(x)$ which converges to $f(x)$ and where $f_{n}(x)$ is in $C^{k}[a,b]$ but $f(x)$ is not.

I thought maybe $f_{n}(x)=x^{k+1}sin(\frac{1}{nx})$ would do it since it's not k+1 differentiable at 0. But then $f(x)=0$ which can be differentiated infinitely (since each derivative is 0).

2. Apr 12, 2012

clamtrox

You are definitely on right track. Try writing your sequence as a series
$f_n = \sum_{i=0}^n a_n \sin(b_n x)$ and then choose an and bn so that limit of fn exists but f'n diverges.

... Or just google "Weierstrass function", if you're lazy :)

3. Apr 12, 2012

micromass

If you're asking such questions, then you should always say which metric you're working with.