Let $p\geq1$ and $s\in(0,1)$. We define the fractional Sobolev space as the space of functions $f\in L^{p}\left(\mathbb{R}\right)$ such that $$\int_{\mathbb{R}}\int_{\mathbb{R}}\left|\frac{f\left(x\right)-f\left(y\right)}{x-y}\right|^{p}\frac{1}{\left|x-y\right|^{1+(s-1)p}}dxdy<+\infty.\tag{1}$$ My question is about the “regularity” of $f$ that can be deduced from $(1)$. In particular, we know that $W^{1,p}\left(\mathbb{R}\right)\subset W^{s,p}\left(\mathbb{R}\right)$, and so $f$ could not be weakly differentiable, but it seems to me that some type of “Lipschitz continuity-type” property must holds. I think this because $(1)$ is bigger than $$\int_{\mathbb{R}}\int_{\left|z\right|<1}\left|\frac{f\left(y+z\right)-f\left(y\right)}{z}\right|^{p}\frac{1}{\left|z\right|^{1+(s-1)p}}dzdy$$ and so some type of regularity must holds for keeping this integral convergent. Maybe I'm completely wrong since I am not able to find articles or books that consider this aspect.
My question is: Am I wrong? What kind of “regularity” we can deduce about a function $f\in W^{s,p}\left(\mathbb{R}\right)$?