Consider the Euclidean space Rn with the orthogonal action of a compact Lie group G. We prove that a locally Lipschitz G-invariant mapping f from Rn to R can be uniformly approximated by G-invariant smooth mappings g in such a way that the gradient of g is a graph approximation of Clarke’s generalized gradient of f. This result enables a proper development of equivariant gradient degree theory for a class of set-valued gradient mappings.

The notion of homotopy in the category of morphisms introduced by Górniewicz and Granas is proved to be equivalence relation which was not clear for years. Some simple properties are proved and a coincidence point index is described.