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Riemann problems at geometric discontinuities are a classic and fascinating issue of hydraulics. In the present paper, the complete solution to the Riemann problem of the one-dimensional Shallow water Equations at monotonic width discontinuities is presented. This solution is based on the assumption that the relationship between the states immediately to the left and to the right of the discontinuity is a stationary weak solution of the one-dimensional variable-width Shallow water Equations. It is demonstrated that the solution to the Riemann problem always exists and it is unique, but there are cases where three solutions are possible. The appearance of multiple solutions is connected to a phenomenon, known as hydraulic hysteresis, observed for supercritical flow in contracting channel. The analysis of a Finite Volume numerical scheme from the literature (Cozzolino et al. 2018b) shows that the algorithm captures the solution with supercritical flow through the width discontinuity when multiple solutions are possible. Interestingly, the one-dimensional variable-width Shallow water Equations are formally identical to the one-dimensional Porous Shallow water Equations, implying that the exact solutions and the numerical scheme discussed in the present paper are relevant for two-dimensional Porous Shallow water numerical models aiming at urban flooding simulations. The exact solution presented here may be used not only as a benchmark, but also as a guide for the construction of new algorithms, and it can be even embedded in an exact solver.