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Global weak continuity of M-functionals in a neighbourhood of the parametric distribution is established. This has implications for robustness of M-estimators vis a vis definitions put forward by Hampel. For instance the Tukey bisquare location estimator is robust on neighbourhoods of the parametric model, but the median is not. 2000 Mathematics subject classification: primary 62G35; secondary 62E20.doi:10.1017/s144678870000149x fatcat:r6xty7rsczdfzjmgalsx73shme