In this paper we analyse functions in Besov spaces Bq,infinity 1/q(RN,Rd),q is an element of(1,infinity)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B<^>{1/q}_{q,\infty }(\mathbb {R}<^>N,\mathbb {R}<^>d),q\in (1,\infty )$$\end{document}, and functions in fractional Sobolev spaces Wr,q(RN,Rd),r is an element of(0,1),q is an element of[1,infinity)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W<^>{r,q}(\mathbb {R}<^>N,\mathbb {R}<^>d),r\in (0,1),q\in [1,\infty )$$\end{document}. We prove for Besov functions u is an element of Bq,infinity 1/q(RN,Rd)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u\in B<^>{1/q}_{q,\infty }(\mathbb {R}<^>N,\mathbb {R}<^>d)$$\end{document} the summability of the difference between one-sided approximate limits in power q, |u+-u-|q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|u<^>+-u<^>-|<^>q$$\end{document}, along the jump set Ju\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {J}_u$$\end{document} of u with respect to Hausdorff measure HN-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {H}<^>{N-1}$$\end{document}, and establish the best bound from above on the integral integral Ju|u+-u-|qdHN-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int _{\mathcal {J}_u}|u<^>+-u<^>-|<^>qd\mathcal {H}<^>{N-1}$$\end{document} in terms of Besov constants. We show for functions u is an element of Bq,infinity 1/q(RN,Rd),q is an element of(1,infinity)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u\in B<^>{1/q}_{q,\infty }(\mathbb {R}<^>N,\mathbb {R}<^>d),q\in (1,\infty )$$\end{document} that lim inf epsilon -> 0+1 epsilon N integral B epsilon(x)|u(z)-uB epsilon(x)|qdz=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \liminf \limits _{\varepsilon \rightarrow 0<^>+}\frac{1}{\varepsilon <^>N}\int _{B_{\varepsilon }(x)} |u(z)-u_{B_{\varepsilon }(x)}|<^>qdz=0 \end{aligned}$$\end{document}for every x outside of a HN-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {H}<^>{N-1}$$\end{document}-sigma finite set. For fractional Sobolev functions u is an element of Wr,q(RN,Rd)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u\in W<^>{r,q}(\mathbb {R}<^>N,\mathbb {R}<^>d)$$\end{document} we prove that lim epsilon -> 0+1 epsilon N integral B epsilon(x)1 epsilon N integral B epsilon(x)|u(z)-u(y)|qdzdy=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim _{\varepsilon \rightarrow 0<^>+}\frac{1}{\varepsilon <^>N}\int _{B_{\varepsilon }(x)}\frac{1}{\varepsilon <^>N}\int _{B_{\varepsilon }(x)} |u\big (z\big )-u(y)|<^>qdzdy=0 \end{aligned}$$\end{document}for HN-rq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {H}<^>{N-rq}$$\end{document} a.e. x, where q is an element of[1,infinity)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q\in [1,\infty )$$\end{document}, r is an element of(0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r\in (0,1)$$\end{document} and rq <= N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$rq\le N$$\end{document}. We prove for u is an element of W1,q(RN),1<q <= N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u\in W<^>{1,q}(\mathbb {R}<^>N),1<q\le N$$\end{document}, that lim epsilon -> 0+1 epsilon N integral B epsilon(x)|u(z)-uB epsilon(x)|qdz=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim \limits _{\varepsilon \rightarrow 0<^>+}\frac{1}{\varepsilon <^>N}\int _{B_{\varepsilon }(x)} |u(z)-u_{B_{\varepsilon }(x)}|<^>qdz=0 \end{aligned}$$\end{document}for HN-q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {H}<^>{N-q}$$\end{document} a.e. x is an element of RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\in \mathbb {R}<^>N$$\end{document}. In addition, we prove Lusin-type approximation for fractional Sobolev functions u is an element of Wr,q(RN,Rd)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u\in W<^>{r,q}(\mathbb {R}<^>N,\mathbb {R}<^>d)$$\end{document} by Holder continuous functions in C0,r(RN,Rd)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C<^>{0,r}(\mathbb {R}<^>N,\mathbb {R}<^>d)$$\end{document}.
