Consideration is given to the approximation in which an arbitrary potential is replaced by a separable potential. A method is presented which permits the construction of a rank-$N$ separable potential which has the property that the resulting $T$ matrix is exact on the energy shell and half off the energy shell at $N$ selected bound state and/or continuum energies. This construction yields a $T$ matrix that is correct off the energy shell in the vicinity of the $N$ on-shell points.