  We consider the problems of sufficiency of jets relative to a given closed
set. In the non-relative case, criteria for r-jets to be V-sufficient and
C^0-sufficient in C^r mappings or C^{r+1} mappings have been obtained. In
particular, it is shown that V-sufficiency and C^0-sufficiency in C^r functions
or C^{r+1} functions are equivalent. In this paper we discuss characterisations
of V-sufficiency and C^0-sufficiency in the relative case, corresponding to the
above non-relative results. Applying the results obtained in the relative case,
we construct examples of polynomial functions whose relative r-jets are
V-sufficient in C^r functions and C^{r+1} functions but not C^0-sufficient in
C^r functions and C^{r+1} functions, respectively. In addition, we give
characterisations of relative finite V-determinacy and also relative finite C^r
contact determinacy.
