Let $W^{k,p}(\mathbb{R}^n)$ be the Sobolev space of $k$ times weakly differentiable functions on $\mathbb{R}^n$, having finite Sobolev $p$ norm. Suppose $\{f_i\}$ is a Cauchy sequence in $W^{k,p}(\mathbb{R}^n)$, such that $f_i: \mathbb{R}^n \rightarrow \mathbb{R}$ is a smooth ($\mathcal{C}^{\infty} $) function for all $i$. Since $W^{k,p}(\mathbb{R}^n)$ is a Banach space (i.e., complete) this Cauchy sequence converges to some function $f$. My question is, what can one say about this function $f$?
Is it continuous? Is it differentiable? Can one conclude any of these facts under some additional hypothesis? Under what hypothesis can one say that the function $f$ has $k$ derivatives (the actual derivative, not the weak derivative)?
Secondly, is it possible to conclude under any reasonable hypothesis that the sequence $f_i$ converges to $f$ uniformly?