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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