Let $\mathcal{D}$ be the subspace of $C[0,1]$ (with uniform metric) consisting of the continuous functions $[0,1]\to \mathbb{R}$ that are differentiable on $(0,1)$. Is $\mathcal{D}$ complete?
I'm thinking no, because I'm thinking that $\mathcal{D}$ is not closed, but I don't know how to show this.
Could somebody please let me know whether I'm right in my reasoning, and if so, how to show this?
If I'm not correct in my reasoning, then if you could please steer me in the right direction, I would very much appreciate it.
Hint: Consider $f_n(x) = |x-1/2|^{1+1/n}, \,n = 1,2,\dots$