Density of test functions in Sobolev space in $\mathbb{R}^n \setminus \{ 0 \}$ Reading my professor's lecture notes on Sobolev spaces I came up with the following proposition: " $\mathcal{D}(\mathbb{R}^n \setminus \{ 0 \})$ is dense in $W^{m,p}(\mathbb{R}^n \setminus \{ 0 \})$ whenever $p \neq \infty$ and $ mp \le n $ thanks to Sobolev-Morrey embeddings". I basically have two issues: first of all, shouldn't Morrey's embedding of a Sobolev space into $C(\mathbb{R}^n)$ require something like $mp > n $ (which is exactly the contrary!)? Furthermore, suppose we have the Morrey embedding of $  W^{m,p}(\mathbb{R}^n \setminus \{ 0 \}) $ into $C(\mathbb{R}^n \setminus \{ 0\})$,  we should have by definition:
$$ \text{cl}(\mathcal{D}(\mathbb{R}^n \setminus \{ 0 \})) = W^{m,p}_0(\mathbb{R}^n \setminus \{ 0 \}) $$
where the closure is taken with respect to the Sobolev space topology. If everything is consistent then we shall conclude:
$$  W^{m,p}_0(\mathbb{R}^n \setminus \{ 0 \})=W^{m,p}(\mathbb{R}^n \setminus \{ 0 \})$$
which sounds a little strange to me.
Now let's talk about norms. Fix $n=1$ for simplicity and suppose that we are able to approximate functions in $W^{m,p}(\mathbb{R} \setminus \{ 0 \})$ with test functions supported in $\mathbb{R} \setminus \{  0\}$. Take $u = H(x) \in W^{m,p}(\mathbb{R} \setminus \{ 0 \})  $ and $(u_n)_n \subset \mathcal{D}(\mathbb{R} \setminus \{ 0 \})$  such that $ \| u_n - u \|_{m,p} \rightarrow 0 $. Clearly such test functions must vanish near the origin forcing the derivatives, and hence their $(m,p)$ norms, to blow up. Am I missing something? Thanks in advance for any comment, suggestion or explanation.
 A: I don't think this is true. If $n=1$ and $g\in W^{1,1}(\mathbb{R}\setminus{0})$ then $g$ has a representative $f$ which is absolutely continuous. By the fundamental theorem of calculus
$$f(x)=f(y)+\int_y^xf'(t)\,dt.$$
Integrate over $(0,1]$ and you get
$$|f(x)|\le \int_0^1|f(y)|\,dy+\int_0^1|f'(t)|\,dt$$
for all $x\in (0,1]$. Hence,
$$\sup_{x\in (0,1]}|f(x)|\le
 \int_0^1|f(y)|\,dy+\int_0^1|f'(t)|\,dt.$$
If you could approximate $g$ in $W^{1,1}(\mathbb{R}\setminus{0})$ with a sequence of functions $g_n$ in $C^\infty_c(\mathbb{R}\setminus{0})$, then by replacing $f$ with $f-g_n$ (which is absolutely continuous) in the previous inequality you would get 
$$\sup_{x\in (0,1]}|f(x)-g_n(x)|\le
 \int_0^1|f(y)-g_n(y)|\,dy+\int_0^1|f'(t)-g_n'(t)|\,dt\to 0.$$
So $g_n\to f$ uniformly. Since $f$ is absolutely continuous, it has a limit $\ell$ as $x\to 0$. If this limit is $\ell\ne 0$, then by uniform convergence $|g_n(x)|\ge\frac{\ell}2$ for all $x\in (0,\delta)$ for some $\delta$ small and all $n$ large.
Update
In the book Tartar there is a chapter called Proving that a point is too small, where he gives the proof for $H^1(\Omega)$ and $n\ge 2$. The proof is too long to write here.
