In this paper, we estimate the area of the graph of a map u:Ω⊂R2→R2\documentclass[12pt]{minimal}
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\begin{document}$${\mathbf{u}}: \varOmega \subset \mathbb {R}^2\rightarrow \mathbb {R}^2$$\end{document} discontinuous on a segment Ju\documentclass[12pt]{minimal}
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\begin{document}$$J_{\mathbf{u}}$$\end{document}, with Ju\documentclass[12pt]{minimal}
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\begin{document}$$J_{\mathbf{u}}$$\end{document} either compactly contained in the bounded open set Ω\documentclass[12pt]{minimal}
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\begin{document}$$\Omega $$\end{document}, or starting and ending on ∂Ω\documentclass[12pt]{minimal}
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\begin{document}$$\partial \Omega $$\end{document}. We characterize A¯∞(u,Ω)\documentclass[12pt]{minimal}
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\begin{document}$$\overline{{\mathcal {A}}}^\infty ({\mathbf{u}},\varOmega )$$\end{document}, the relaxed area functional in a sort of uniform convergence, in terms of the infimum of the area of those surfaces in R3\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {R}^3$$\end{document} spanning the graphs of the traces of u\documentclass[12pt]{minimal}
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\begin{document}$${\mathbf{u}}$$\end{document} on the two sides of Ju\documentclass[12pt]{minimal}
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\begin{document}$$J_{\mathbf{u}}$$\end{document} and having what we have called a semicartesian structure. We exhibit examples showing that A¯(u,Ω)\documentclass[12pt]{minimal}
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\begin{document}$$\overline{{\mathcal {A}}}({\mathbf{u}},\varOmega )$$\end{document}, the relaxed area in L1(Ω;R2)\documentclass[12pt]{minimal}
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\begin{document}$$L^1(\varOmega ; \mathbb {R}^2)$$\end{document}, may depend on the values of u\documentclass[12pt]{minimal}
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\begin{document}$${\mathbf{u}}$$\end{document} far from Ju\documentclass[12pt]{minimal}
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\begin{document}$$J_{\mathbf{u}}$$\end{document} and also on the relative position of Ju\documentclass[12pt]{minimal}
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\begin{document}$$J_{\mathbf{u}}$$\end{document} with respect to ∂Ω\documentclass[12pt]{minimal}
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\begin{document}$$\partial \varOmega $$\end{document}. These examples confirm the highly non-local behavior of A¯(u,·)\documentclass[12pt]{minimal}
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\begin{document}$$\overline{{\mathcal {A}}}({\mathbf{u}},\cdot )$$\end{document} and justify the interest in the study of A¯∞\documentclass[12pt]{minimal}
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\begin{document}$$\overline{{\mathcal {A}}}^\infty $$\end{document}. Finally we prove that A¯(u,·)\documentclass[12pt]{minimal}
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\begin{document}$$\overline{{\mathcal {A}}}({\mathbf{u}},\cdot )$$\end{document} is not subadditive for a rather large class of discontinuous maps u\documentclass[12pt]{minimal}
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\begin{document}$${\mathbf{u}}$$\end{document}.
