# The gradient's limit of a definable family of functions admits a variational stratification

arXiv: https://arxiv.org/abs/2402.08272 · HTML: https://mlsanity.com/paper/2402.08272/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Feb 2024.

## Built on by (most cited first, 2 of 2)

- [Convergence of empirical subgradients for optimal transport-based objectives](https://mlsanity.com/paper/2605.28134/) (2026)
- [Lipschitzian SLLNs for random functions](https://mlsanity.com/paper/2607.20411/) (2026)

## Abstract (arXiv)

It is well-known that the convergence of a family of smooth functions does not imply the convergence of its gradients. In this work, we show that if the family is definable in an o-minimal structure (for instance semialgebraic, subanalytic, or any composition of the previous with exp, log), then the gradient's limit admits a variational stratification and, under mild assumptions, is a conservative set-valued field in the sense introduced by Bolte and Pauwels. Immediate implications of this result on convergence guarantees of smoothing methods are discussed. The result is established in a general form, where the functions in the original family might be non Lipschitz continuous, be vector-valued and the gradients are replaced by their Clarke Jacobians or an arbitrary mapping satisfying a definable variational stratification. In passing, we investigate various stability properties of definable variational stratifications which might be of independent interest.