Let $M,N$ be smooth $d$-dimensional Riemannian manifolds.
A Sobolev map $f \in W^{1,p}(M,N)$ is called localizable if for every $x_0 \in M$ there exists a neighbourhood $U$ of $x_0$ in $M$ and a domain $V$ of a coordinate chart in $N$ with the property that $f(U) \subseteq V$.
Question: For which $p$ every Sobolev map in $W^{1,p}(M,N)$ is localizable? I know that for $p>d$ every Sobolev map is continuous, hence in particular localizable. Is there anythig general we can say about $r<p <d$ for some $r>1$?
I am in particularly interested in $p=\frac{d}{2}$. Is it true that every Sobolev space in $W^{1,\frac{d}{2}}(M,N)$ localizable?