The problem of the identifiability of the mixing distribution and of the unknown parameters for a continuous mixture of Gaussian distributions is considered. Relevance of the problem under various analytical, statistical, and applicative points of view is stressed. Uniqueness of the mixing distribution and of the mean and variance functions for the mixed Gaussian distribution is proved. Furthermore, their continuous dependence on the mixture itself is proved under suitable topologies. These results also extend to the multidimensional case and to the case of non-Gaussian distributions, and/or signed mixing measure.