We obtain approximation results for general positive linear operators satisfying mild conditions, when acting on discontinuous functions and absolutely continuous functions having discontinuous derivatives. The upper bounds, given in terms of a local first modulus of continuity, are best possible, in the sense that we can construct particular sequences of operators attaining them. When applied to functions of bounded variation or absolutely continuous functions having derivatives of bounded variation, these upper bounds are better and simpler to compute than the usual total variation bounds. The particular case of the Bernstein polynomials is thoroughly discussed. We use a probabilistic approach based on representations of such operators in terms of expectations of random variables.