We present an approximation result for functions u:Ω→Rn\documentclass[12pt]{minimal}
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\begin{document}$$u:\Omega \rightarrow \mathbb {R}^n$$\end{document} belonging to the space GSBD(Ω)∩L2(Ω,Rn)\documentclass[12pt]{minimal}
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\begin{document}$$GSBD(\Omega )\cap L^2(\Omega ,{{\mathbb R}}^n)$$\end{document} with e(u)\documentclass[12pt]{minimal}
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\begin{document}$$e(u)$$\end{document} square integrable and Hn-1(Ju)\documentclass[12pt]{minimal}
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\begin{document}$$\fancyscript{H}^{n-1}(J_u)$$\end{document} finite. The approximating functions uk\documentclass[12pt]{minimal}
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\begin{document}$$u_k$$\end{document} are piecewise continuous functions such that uk→u\documentclass[12pt]{minimal}
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\begin{document}$$u_k\rightarrow u$$\end{document} in L2(Ω,Rn)\documentclass[12pt]{minimal}
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\begin{document}$$L^2(\Omega ,\mathbb {R}^n)$$\end{document}, e(uk)→e(u)\documentclass[12pt]{minimal}
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\begin{document}$$e(u_k)\rightarrow e(u)$$\end{document} in L2(Ω,Msymn×n)\documentclass[12pt]{minimal}
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\begin{document}$$L^2(\Omega ,\mathbb {M}^{n{\times }n}_{sym})$$\end{document}, Hn-1(Juk▵Ju)→0\documentclass[12pt]{minimal}
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\begin{document}$$\fancyscript{H}^{n-1}(J_{u_k}\triangle J_u)\rightarrow 0$$\end{document}, and ∫Juk∪Ju|uk±-u±|∧1dHn-1→0\documentclass[12pt]{minimal}
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\begin{document}$$\int _{J_{u_k}\cup J_u}|u_k^\pm -u^\pm |\wedge 1d{\fancyscript{H}}^{n-1}\rightarrow 0$$\end{document}. As an application, we provide the extension to the vector-valued case of the Γ\documentclass[12pt]{minimal}
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\begin{document}$$\Gamma $$\end{document}-convergence result in GSBV(Ω)\documentclass[12pt]{minimal}
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\begin{document}$$GSBV(\Omega )$$\end{document} proved by Ambrosio and Tortorelli (Commun Pure Appl Math 43:999–1036, 1990; Boll. Un. Mat. Ital. B (7) 6:105–123, 1992).
