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In this paper we describe the notion of a weak lipschitzianity of a mappingon a $C^{q}$ stratification. We also distinguish a class of regularityconditions that are in some sense invariant under definable, locally Lipschitzand weakly bi-Lipschitz homeomorphisms. This class includes the Whitney (B)condition and the Verdier condition.

Invariance of regularity conditions under definable, locally Lipschitz, weakly bi-Lipschitz mappings