Considering gravitational collapse of Type I matter fields, we prove that, given an arbitrary C2-mass functionM(r, v) and a C1-functionh(r, v) (through the corresponding C1-metric functionν(t, r)), there exist infinitely many choices of energy distribution functionb(r) such that the ’true’ initial data(M, h(r,v)) leads the collapse to the formation of naked singularity. We further prove that the occurrence of such a naked singularity is stable with respect to small changes in the initial data. We remark that though the initial data leading to both black hole (BH) and naked singularity (NS) form a ’big’ subset of the true initial data set, their occurrence is not generic. The terms ’stability’ and ’genericity’ are appropriately defined following the theory of dynamical systems. The particular case of radial pressurepr(r) has been illustrated in details to get a clear picture of how naked singularity is formed and how, it is stable with respect to initial data.