The gradient's limit of a definable family of functions is aconservative set-valued field

It is well-known that the convergence of a family of smooth functions does not implythe convergence of its gradients. In this work, we show that if the family is definablein an o-minimal structure (for instance semialgebraic, subanalytic, or any compositionof the previous with exp, log), then the gradient's limit is a conservative set-valued fieldin the sense introduced by Bolte and Pauwels. Immediate implications of this resulton convergence guarantees of smoothing methods are discussed. Finally, a more generalresult is established, where the functions in the original family might be merely Lipschitzcontinuous, vector-valued and the gradients are replaced by their Clarke's Jacobians oran arbitrary definable conservative mapping.