Let \(\mathcal D\) be a domain of holomorphy in \(\mathbb C^n\) and \(S\) a closed subset of \(\mathcal D\), dividing \(\mathcal D\) into two subdomains \(\mathcal D^{\pm}\). The authors assume that \(S=\mathcal M\cup\sigma\), where \(\sigma\) is a closed subset of zero \((2n-1)\)-dimensional measure and \(\mathcal M\) is a \(\mathcal C^1\)-smooth oriented hypersurface in \(\mathcal D\setminus\sigma\). They investigate the possibility of representing a CR function \(f\), defined and \(L^1_{\text{loc}}\) in \(\mathcal M\), as the jump of two holomorphic functions \(\widetilde f^{\pm}\) defined in \(\mathcal D^{\pm}\), and show that this is possible when \(f\) satisfies suitable conditions near the singular set \(\sigma\). They also investigate the behaviour near \(\sigma\) of the representing functions \(\widetilde f^{\pm}\) in the case that it consists of simple singularities like cuspidal points, edges, corners.