We put, Sobolev space $$H_{s}=H_{s}(\mathbb R)=\{f\in L^{2}(\mathbb R):[\int_{\mathbb R} |\hat{f}(\xi)|^{2}(1+|\xi|^{2})^{s}d\xi]^{1/2}<\infty \}.$$ If $U$ is an open set in $\mathbb R,$ the localized Sobolev space $H_{s}^{loc}$ is the set of all distributions $f\in \mathcal{D'}(U)$ such that for every precompact open set $V$ with $\overline{V}\subset U$ there exists $g\in H_{s}$ such that $g=f$ on $V.$
Fact. A distribution $f\in \mathcal{D'}$ is in $H_{s}^{loc}(U)$ iff $\phi f \in H_{s}$ for every $\phi \in C_{c}^{\infty}(U).$ (where, $C_{c}^{\infty}(U)=$ The space space of $C^{\infty}$ functions whose support is compact)
We put, $C(\mathbb R)=$ The space of continuous functions on $\mathbb R.$
My Question is: Let $s>3/2.$ Is it true that $C(\mathbb R)\subset H_{s}^{loc}$ ? If not, for which $s>0,$ one can expect this ?
Functions in $H_1$ are absolutely continuous (or more precisely have representatives that are absolutely continuous). Since $H_s^{loc} \subset H_1^{loc}$ for $s > 1$ an affirmative answer to your question would imply that functions in $C(\mathbb R)$ have AC representatives, which they do not.