Banach spaces, $C^1([-1,1])$

Question

Denote by $C^1([a,b])$ the vector space of continuously differentiable functions with derivatives at $a,b$ defined as one-sided limits. I was wondering whether the linear subspace $C^1([-1,1])$ of $C^0([-1,1])$ closed with respect to $||\cdot||_{C^0}$?

I thought it was not, and I wanted to prove that $|x|$ is a limit point of $C^1([-1,1])$ implying $C^1([-1,1])$ is not closed. However, I have troubles finding a sequence of differentiable functions which converges to $|x|$. Any suggestions would be appreciated!

Answer 1

Hint: If $x\in[-1,1]$, what is $\displaystyle\lim_{n\to\infty}\sqrt{x^2+\frac1{n^2}}$?

Answer 2

It's not closed, which is a result of the Weierstrass Approxmation Theorem: Given any $f\in C^0[-1,1]$ and any $\epsilon >0$, there is a polynomial $p$ such that $\|f-p\|_{C^0} < \epsilon$. In fact this shows that $C^\infty$ is not a closed subspace of $C^0$ since polynomials are smooth.