In this paper, we studied continuous radial functions that are positive definite on a domain D in the Euclidean space Rd or a compact two-point homogeneous space Md. We showed that for D?Rd that contains d-balls of arbitrary radius, a radial function that is PD on D is PD on Rd. On the other hand, for any closed proper subset D?Md, there exists a radial function that is PD on D but not PD on Md. We derived some sufficient conditions in terms of spectral coefficients for a continuous radial function that is PD on D to be PD on Md. As an example, we explicitly constructed radial functions that are PD on the unit ball embedded in the unit sphere Sd by a distance preserving map, but not PD on Sd.