<p>This study explores a broader variety of single-index multiplicative models (SIMM for short) with an unknown, discontinuous link function. Relaxing the continuity assumption in nonparametric functions enhances the applicability to positive data. However, the authors find an issue with the existing least product relative error (LPRE for short) technique at jump points, posing a challenge in estimating the link function accurately in SIMMs. They propose an automated method that combines the LPRE technique with jump-preserving methods to simultaneously estimate the unknown parameter vector and the discontinuous link function. Their approach is flexible and practical, not requiring prior knowledge of jump point details. Furthermore, they establish the asymptotic properties of the estimators for the parametric vector and the discontinuous function components under reasonable conditions. To validate their approach, they conduct numerical simulations evaluating the performance with finite samples. Additionally, they demonstrate the effectiveness of the approach through real data analysis.</p>