This is exercise 3.11 of the book $Introduction\ to\ Nonlinear\ Dispersive\ Equations$ written by Felipe Linares and Gustavo Ponce.
Let's define the fractional sobolev space with $0<s<1,\ p\in (1,\infty)$,
$L^p_s(\mathbb{R}^n):=\{f\in L^p(\mathbb{R}^n):D^sf\in L^p(\mathbb{R^n})\}$, where the fractional derivative $\hat{(D^sf)} := (2\pi i|\xi|)^s\hat{f}$ and ^ denotes the fourier transform.
Now we assume $n=1$ and $s=\frac{1}{p}$, can we prove that any function in $L^p(\mathbb{R})$ with (at least) a jump discontinuity does not belong to $L^p_{\frac{1}{p}}(\mathbb{R})$ ? Here, a jump point $x_0$ means a point where $f(x_0-)$ and $f(x_0+)$ both exist and are both finite, but $f(x_0-)\neq f(x_0+)$.
I think this is a critical case where $n=sp$. Since we already know that if $n<sp$ , the function itself has a continuous version and thus has no jump discontinuity.
