The linear spectrum of a non-homogeneous, compressible and thermally conducting planar plasma with a vertical gravitational field and a sheared horizontal magnetic field is studied. It is shown that the spectrum has two continuous parts. The Alfvén continuum of linear ideal MHD is unaffected by radiative conduction, but the slow continuum is removed and replaced by a new continuous part which has been called isothermic continuum. Purely exponentially growing and overstable normal modes have been determined numerically and the distribution of their eigenfrequencies in the complex plane has been studied. The eigenfrequencies lie on specific curves in the complex plane which are partially determined by the isothermic continuum.