Spatiotemporal correlations of the one-dimensional spring-block (Burridge-Knopoff) model of earthquakes, either with or without the viscosity term, are studied by means of numerical computer simulations. The continuum limit of the model is examined by systematically investigating the model properties by varying the block-size parameter a toward a → 0. The Kelvin viscosity term is introduced so that the model dynamics possesses a sensible continuum limit. In the presence of the viscosity term, many of the properties of the original discrete BK model are kept qualitatively unchanged even in the continuum limit, although the size of minimum earthquakes gets smaller as a gets smaller. One notable exception is the existence/nonexistence of the doughnutlike quiescence prior to the mainshock. Although large events of the original discrete BK model accompany seismic acceleration together with a doughnutlike quiescence just before the mainshock, the spatial range of the doughnutlike quiescence becomes narrower as a gets smaller, and in the continuum limit, the doughnutlike quiescence might vanish altogether. The doughnutlike quiescence observed in the discrete BK model is then a phenomenon closely related to the short-length cutoff scale of the model.