A linear stability analysis of a shear flow in the presence of a continuous but steep variation of viscosity between two layers of nearly uniform viscosity is presented. This instability is investigated in relation to the known interfacial instability for the parallel flow of two superposed fluids of different viscosity. With respect to this configuration, the stability of our problem depends on two new parameters: the interface thickness $\delta$ and the Péclet number $\hbox{\it Pe}$, which accounts for diffusion effects when viscosity perturbations, coupled to the velocity perturbations, are allowed. We show that instability still exists for the continuous viscosity profile, provided the thickness of the interface is small enough and $\hbox{\it Pe}$ sufficiently large. Small and large wavenumbers are found to be stable, at variance with the discontinuous configuration. Of particular interest is also the possibility of obtaining higher growth rates than in the discontinuous case for suitable $\hbox{\it Pe}$ and $\delta$ ranges.