We formulate a discrete version of time-dependent DFT well suited to study model systems and we then apply it to the Anderson model. We show that the derivative discontinuity of the exchange-correlation density functional is intimately related to the Coulomb blockade phenomenon. To include the discontinuity we propose an approximate functional based on finite-temperature DFT. This functional yields the exact Kohn-Sham potential at the particle-hole symmetric point and exhibits a derivative discontinuity in the limit of zero temperature. We then use it to study the zero-temperature conductance within the standard Landauer formalism and show that the Kondo plateau is accurately reproduced. On the other hand, at the Kondo temperature the exact Kohn-Sham conductance overestimates the exact one by an order of magnitude. To understand the failure of DFT we resort to its time-dependent version and conclude that the suppression of the Kondo resonance with increasing temperature must be attributed to dynamical exchange-correlation corrections.