Potential functional approximations are an intriguing alternative to density functional approximations. The potential functional that is dual to the Lieb density functional is defined and its properties are reported. The relationship between the Thomas-Fermi theory as a density functional and the theory as a potential functional is derived. The properties of several recent semiclassical potential functionals are explored, especially regarding their approach to the large particle number and classical continuum limits. The lack of ambiguity in the energy density of potential functional approximations is demonstrated. The density-density response function of the semiclassical approximation is calculated and shown to violate a key symmetry condition.