In this paper, we study diagonalizable hyperbolic systems in one space dimension. Based on a new gradient entropy estimate, we prove the global existence of a continuous solution, for large and nondecreasing initial data. Moreover, we show in particular cases some uniqueness results. We also remark that these results cover the case of systems which are hyperbolic but not strictly hyperbolic. Physically, this kind of diagonalizable hyperbolic systems appears naturally in the modelling of the dynamics of dislocation densities.