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Diverse numerical studies of the 1D Coulomb problem have suggested the existence of an in nite binding-energy ground-state and de nite parity eigenstates in this system. We exhibit that such conclusions about the 1D Coulomb problem are wrong. The origin of the mistakes are to be found not in the numerical techniques employed but in the careless extrapolation of their results. Any numerical technique used for solving the 1D Coulomb problem must be supplemented by a detailed analysis of the e ect of the impenetrable barrier produced by the  jxj ¹ singularity in order to obtain valid conclusions.