SUMMARY The necessary and sufficient conditions for the existence of a solution to the one-dimensional magnetotelluric inverse problem, are obtained in the form of a concise formula, which involves only the finite collection of response data values. These conditions are easy to apply in practice since they only involve testing whether the two Hermitian forms QN(ζ) and QN(ζ) are non-negative or, equivalently, testing whether the two associated (N x N) Hermitian matrices QN, and QN are non-negative definite. These conditions are obtained using the Nevanlinna-Pick theorem, after an appropriate reformulation of the existence result due to Parker. The present approach enables one to decide whether or not a given response data set is singular, by an examination of the ranks of QN and QN. Moreover, the ranks of these two matrices enable the a priori deduction of the form and degree of a rational, interpolating response function that is compatible with the data set. In addition, it is shown how this rational function may be constructed directly from the given data set. Since a response function is rational if and only if it arises from a conductivity distribution consisting of a stack of thin conducting sheets (σɛD+), the theory clarifies the important role played by this class of conductivity profiles in the characterization of the existence result. In particular, it is shown that for a P1(2)-admissible response data set for which QN≥ON, and QN≥ON, it is always possible to characterize the data with a σɛD+ consisting of exactly v conducting sheets, where v= rank (QN). In addition, this characterization is unique if and only if either QN, and/or QN are singular. Also, it is shown how to extrapolate a P1(2)-admissible finite data set by the computation of a region of P1(2)-admissibility; if the response is constrained to lie in this region, then the extrapolated data set is always compatible with some one-dimensional profile.