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The connection between the scaling function, directly extracted from the analysis of electron scattering data, and the nuclear spectral function or nuclear momentum density is investigated at depth. The dependence of the scaling function on the two independent variables in the scattering process, the transfer momentum ($q$) and the scaling variable ($y$), is taken into account, and the analysis is extended to both, positive and negative $y$-values, i.e., below and above the center of the quasielastic peak, respectively. Analytical expressions for the derivatives of the scaling function, evaluated at the finite limits of integration dealing with the kinematically allowed region, are connected with the spectral function. Here, contributions corresponding to zero and finite excitation energies are included. The scaling function is described by the Gumbel density distribution, whereas short-range correlations are incorporated in the spectral function by using some simple models. Also different parametrizations for the nucleon momentum distribution, that are compatible with the general properties of the scaling function, have been considered.