We derive, for the case of the asymmetric Hubbard dimer at half-filling, the exact functional associated with each singlet ground and excited state, using both Levy's constrained search and Lieb's convex formulation. While the ground-state functional is, as commonly known, a convex function with respect to the density (or, more precisely, the site occupation), the functional associated with the (highest) doubly-excited state is found to be concave. Also, we find that, because the density of the first excited state is non-invertible, its "functional" is a partial, multi-valued function composed of one concave and one convex branch that correspond to two separate sets of values of the external potential. These findings offer insight into the challenges of developing state-specific excited-state density functionals for general applications in electronic structure theory.