For a nondecreasing score function having finitely many jump discontinuities, a representation of $M$-estimators with the second-order asymptotic distribution is established, and the result is also extended to one-step versions of $M$-estimators.
Jana Jureckova. Pranab Kumar Sen. "A Second-Order Asymptotic Distributional Representation of $M$-Estimators with Discontinuous Score Functions." Ann. Probab. 15 (2) 814 - 823, April, 1987. https://doi.org/10.1214/aop/1176992174