 Rejection point of an influence function

For observations expressed in centered, standardized coordinates, the rejection point iswith value $\infty$ if the set is empty. A finite rejection point means sufficiently distant contamination has zero first-order effect. Boundedness of the influence function does not imply a finite rejection point: the influence function of a quantile has nonzero constant tails. Nor does tending to zero at infinity guarantee a finite rejection point unless those tails become identically zero.