Recent developments in the dedicated scanning transmission electron microscope (STEM) now permit probe sizes of atomic dimensions to be obtained (0.22 nm FWHM at 100 kV). When coupled with an efficient parallel detector for electron energy-loss spectroscopy (EELS), there now exists the potential to obtain EELS with atomic resolution, This potential is further enhanced by the Z-contrast imaging technique in the STEM, which can produce incoherent, atomic-resolution images of crystalline structures, The Z-contrast image, being formed from only the high-angle thermal diffuse scattering, can thus be acquired simultaneously with the energy-loss spectrum and used to position the probe with atomic precision. However, to obtain an atomic-resolution energy-loss spectrum the range of the inelastic scattering must be less than the atomic spacing. Additionally, to enable a direct correlation between the spectrum and the Z-contrast image, i.e. a description of the spatial resolution of the spectrum in terms of a convolution of a probe intensity profile and an object function, we need to determine the conditions for incoherent imaging with inelastically scattered electrons. In this paper, a theorem for the spatial resolution of the inelastic signal first proposed by Ritchie and Howie and a hydrogenic model for the object function developed by Maslen and Rossouw are combined to calculate the object functions for several characteristic K-edges. More generally, the collection conditions, energy range and detectability for atomic-resolution spectra are discussed, Finally, atomic-resolution spectroscopy is demonstrated with a cobalt L-edge profile of an atomically abrupt CoSi2/Si (111) interface.