Please give, explicitly, a function $f:\Omega\mapsto\mathbb{R}$ such that $f\in W^{k,p}(\Omega)$ but $f\notin W^{s,p}(\Omega)$ for $s>k$.
Here $\Omega$ can be a subset of $\mathbb{R}^n$ with desired boundary, $\mathbb{R}^n$ or torus $\mathbb{T}^n$. 



Examples of functions in $W^{k,p}(\Omega)$ with exact smoothness
If $\Omega=B(0,1)$, take a radial function $f(r)=\frac{1}{r^{n/p}\left(\log r-1\right)^{2/p}}$ Then
$$
\int f^p dx = \omega\_n \int\_0^1 \frac{1}{r\left(\log r-1\right)^2} dr =\omega\_n, 
$$
so $f\in L^p$, but not in $L^q$ for $q>p$, and therefore not in any W^{s,p}, $s>0$.


Now integrate $f$ k times in $r$, say $F\_k(r)=\int\_0^r \int\_0^{t\_1} ..\int\_0^{t\_{k-1}} f(u)du$. Then $F\_k\in W^{k,p}(\Omega)$, but not in $W^{s,p}(\Omega)$ for $s>k$.


In the $p=\infty$ case, choose
$$
f(r)=1 \mbox{ for } r\leq \frac{1}{2} \mbox{ and } 0 \mbox{ otherwise.} 
$$
Then $f$ is bounded but not continuous and therefore not in any $W^{s,p}(\Omega)$ space for $s>0$, $q>1$ and $sq>n$. Then proceed as above.


