We consider one-dimensional linear transport equations with bounded but possibly discontinuous coefficient $a$. The Cauchy problem is studied from two different points of view. In the first case we assume that $a$ is piecewise continuous. We give an existence result and a precise description of the solutions on the lines of discontinuity. In the second case, we assume that $a$ satisfies a one-sided Lipschitz condition. We give existence, uniqueness and general stability results for backward Lipschitz solutions and forward measure solutions, by using a duality method. We prove that the flux associated to these measure solutions is a product by some canonical representative $\widehat{a}$ of $a$.

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