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[Número especial de la revista con las participaciones al congreso "[EFMC8] 8th European Fluid Mechanics Conference"] The viscous linear stability of parallel gaseous jets with piecewise constant base profiles is considered in the limit of low Mach numbers. Our results generalise those of Drazin [P.G. Drazin, Discontinuous velocity profiles for the Orr-Sommerfeld equation J. Fluid Mech. 10 (1961) 571-583], by contemplating the possibility of arbitrary jumps in density and transport properties between two uniform streams separated by a vortex sheet. The eigenfunctions, obtained analytically in the regions of uniform flow, are matched through an appropriate set of jump conditions at the discontinuity of the basic flow, which are derived by repeated integration of the linearised conservation equations in their primitive variable form. The development leads to an algebraic dispersion relation of ample validity that explicitly accounts for the parametric dependence of the stability properties on the jet-to-ambient density ratio, the Reynolds number, the Prandtl number, and the exponent of the presumed power-law dependence of viscosity and thermal conductivity on temperature. The dispersion relation is validated through comparisons with stability calculations performed with continuous profiles and is applied, in particular, to study the effects of molecular transport on the spatiotemporal stability of parallel non-isothermal gaseous jets with very thin shear layers. The eigenvalue computations performed by using the vortex-sheet model are shown to be several orders of magnitude faster than those associated with continuous profiles with thin shear layers.