  In this paper we analyse functions in Besov spaces
$B^{1/q}_{q,\infty}(\mathbb{R}^N,\mathbb{R}^d),q\in (1,\infty)$, and functions
in fractional Sobolev spaces $W^{r,q}(\mathbb{R}^N,\mathbb{R}^d),r\in
(0,1),q\in [1,\infty)$. We prove for Besov functions $u\in
B^{1/q}_{q,\infty}(\mathbb{R}^N,\mathbb{R}^d)$ the summability of the
difference between one-sided approximate limits in power $q$, $|u^+-u^-|^q$,
along the jump set $\mathcal{J}_u$ of $u$ with respect to Hausdorff measure
$\mathcal{H}^{N-1}$, and establish the best bound from above on the integral
$\int_{\mathcal{J}_u}|u^+-u^-|^qd\mathcal{H}^{N-1}$ in terms of Besov
constants. We show for functions $u\in
B^{1/q}_{q,\infty}(\mathbb{R}^N,\mathbb{R}^d),q\in (1,\infty)$ that
\begin{equation} \liminf\limits_{\varepsilon \to 0^+}\fint_{B_{\varepsilon}(x)}
|u(z)-u_{B_{\varepsilon}(x)}|^qdz=0 \end{equation} for every $x$ outside of a
$\mathcal{H}^{N-1}$-sigma finite set. For fractional Sobolev functions $u\in
W^{r,q}(\mathbb{R}^N,\mathbb{R}^d)$ we prove that \begin{equation}
\lim_{\rho\to 0^+}\fint_{B_{\rho}(x)}\fint_{B_{\rho}(x)}
|u\big(z\big)-u(y)|^qdzdy=0 \end{equation} for $\mathcal{H}^{N-rq}$ a.e. $x$,
where $q\in[1,\infty)$, $r\in(0,1)$ and $rq\leq N$. We prove for $u\in
W^{1,q}(\mathbb{R}^N),1<q\leq N$, that \begin{equation}
\lim\limits_{\varepsilon\to 0^+}\fint_{B_{\varepsilon}(x)}
|u(z)-u_{B_{\varepsilon}(x)}|^qdz=0 \end{equation} for $\mathcal{H}^{N-q}$ a.e.
$x\in \mathbb{R}^N$.
